Toric Varieties

David Cox

John Little

Hal Schenck

Department of Mathematics and Computer Science, Amherst College, Amherst, MA 01002

E-mail address: dac@cs.amherst.edu

Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610

E-mail address: little@mathcs.holycross.edu

Department of Mathematics, Texas A&M University, College Station, TX 77843

E-mail address: schenck@math.tamu.edu

Contents

Notation v

Part I: Elementary Theory of Toric Varieties Chapter 1. Affine Toric Varieties 3

§0. Background: Affine Varieties 3 §1. Introduction to Affine Toric Varieties 9 §2. Cones and Affine Toric Varieties 21 §3. Properties of Affine Toric Varieties 32

Chapter 2. Projective Toric Varieties 45 §0. Background: Projective Varieties 45 §1. Lattice Points and Projective Toric Varieties 50 §2. Lattice Points and Polytopes 58 §3. Polytopes and Projective Toric Varieties 69 §4. Properties of Projective Toric Varieties 81

Chapter 3. Normal Toric Varieties 89 §0. Background: Abstract Varieties 89 §1. Fans and Normal Toric Varieties 101 §2. The Orbit–Cone Correspondence 110 §3. Equivariant Maps of Toric Varieties 118 §4. Complete and Proper 127

Chapter 4. Homogeneous Coordinates 135 §0. Background: Quotients in Algebraic Geometry 135 §1. Total Coordinate Rings of Toric Varieties 135 §2. Quotient Constructions of Toric Varieties 135 §3. Examples 136 §4. Polytopes and Homogenization 136

Part II: Further Theory of Toric Varieties

Chapter 5. Divisors on Toric Varieties 139 §1. Normality and Divisors 139 §2. Cartier and Weil Divisors on Toric Varieties 139 §3. Divisor Classes on Toric Varieties 139

Chapter 6. Sheaves on Toric Varieties 141 §1. Invertible and Reflexive Sheaves 141 §2. Connection to Polytopes 141 §3. Ampleness Criteria on Toric Varieties 141 §4. The Canonical Sheaf 142

Chapter 7. Sheaf Cohomology of Toric Varieties 143 §1. Čech Cohomology 143 §2. Vanishing Theorems 143 §3. Serre Duality and Ehrhart polynomials 143

Chapter 8. Toric Surfaces 145 §1. Singularities of Toric Surfaces and Their Resolutions 145 §2. Hirzebruch-Jung Continued Fractions 155 §3. Smooth Toric Surfaces 164 §4. Riemann-Roch and Lattice Polygons 174

Chapter 9. Resolution of Singularities 187

Part III: Advanced Theory of Toric Varieties

Chapter 10. Geometric Invariant Theory 191

Chapter 11. The Riemann-Roch Theorem 193

Chapter 12. The Topology of Toric Varieties 195

Chapter 13. Multigraded Commutative Algebra 197

Bibliography 199

Notation

ℤ, ℚ, ℝ, ℂ Integers, rational numbers, real numbers, complex numbers
ℕ  Semigroup of nonnegative integers {0, 1, 2, … }
ℂ^{∗}  Multiplicative group of nonzero complex numbers ℂ \ {0}
M  Character group of the torus T_{N}
M_{ℝ}, M_{ℚ} Vector spaces M ⊗_{ℤ} ℝ, M ⊗_{ℤ} ℚ built from M
N  Lattice of 1-parameter subgroups of the torus T_{N}
N_{ℝ}, N_{ℚ} Vector spaces N ⊗_{ℤ} ℝ, N ⊗_{ℤ} ℚ built from N
σ  Rational convex polyhedral cone in N_{ℝ}
σ^{∨}  Dual cone of σ
Σ  Fan in N_{ℝ}
Σ(r)  r-dimensional cones of Σ
Σ_{max}  Maximal cones of Σ
Y_{A}  Affine toric variety of a finite set A ⊂ M
X_{A}  Projective toric variety of a finite set A ⊂ M
X_{Σ}  Toric variety of a fan Σ
X_{P}  Projective toric variety of a lattice polytope P
Cl(X)  Divisor class group of a normal variety X
Pic(X)  Picard group of a normal variety X

Part I

Elementary Theory of Toric Varieties

Chapters 1 to 4 introduce the theory of toric varieties. This part of the book assumes only a minimal amount of algebraic geometry, at the level of Ideals, Varieties and Algorithms [7]. Each chapter begins with a background section that develops the necessary algebraic geometry.

Chapter 1

Affine Toric Varieties

0. Background: Affine Varieties

We begin with the algebraic geometry needed for our study of affine toric varieties. Our discussion assumes Chapters 1–5 and 9 of [7].

Coordinate Rings. An ideal I ⊆ S = ℂ[x_{0}, …, x_{n}] gives an affine variety

V(I) = {p ∈ ℂ^{n} | f(p) = 0 for all f ∈ I}

and an affine variety V ⊆ ℂ^{n} gives the ideal

I(V) = {f ∈ S | f(p) = 0 for all p ∈ V }.

By the Hilbert Basis Theorem, an affine variety V is defined by the vanishing of finitely many polynomials in S, and for any ideal I, the Nullstellensatz tells us that I(V(I)) = \sqrt{I} = \{f ∈ S | f^{k} ∈ I for some k ≥ 1\} since ℂ is algebraically closed. The most important algebraic object associated to V is its coordinate ring

ℂ[V] = S/I(V).

Elements of ℂ[V] can be interpreted as the ℂ-valued polynomial functions on V. Note that ℂ[V] is a ℂ-algebra, meaning that its vector space structure is compatible with its ring structure. Here are some basic facts about coordinate rings:

• ℂ[V] is an integral domain ⇔ I(V) is a prime ideal ⇔ V is irreducible.

• Polynomial maps (also called morphisms) F : V_{1} → V_{2} between affine varieties correspond to ℂ-algebra homomorphisms F^{∗} : ℂ[V_{2}] → ℂ[V_{1}], where F^{∗}(g) = g ∘ F for g ∈ ℂ[V_{2}].

• Two affine varieties are isomorphic if and only if their coordinate rings are isomorphic ℂ-algebras.

• A point p of an affine variety V gives the maximal ideal

{f ∈ ℂ[V] | f(p) = 0} ⊆ ℂ[V],

and all maximal ideals of ℂ[V] arise this way. Coordinate rings of affine varieties can be characterized as follows (Exercise 0.1).

Lemma 0.1. A ℂ-algebra R is isomorphic to the coordinate ring of an affine variety if and only if R is a finitely generated ℂ-algebra with no nonzero nilpotents, i.e., if f ∈ R satisfies f^{m} = 0, then f = 0. □

To emphasize the close relation between V and ℂ[V], we sometimes write

(0.1)  V = Spec(ℂ[V]).

This can be made canonical by identifying V with the set of maximal ideals of ℂ[V] via the fourth bullet above. More generally, one can take any commutative ring R and define the affine scheme Spec(R). The general definition of Spec uses all prime ideals of R, not just the maximal ideals as we have done. Thus some authors would write (0.1) as V = Specm(ℂ[V]), the maximal spectrum of ℂ[V]). Readers wishing to learn more about affine schemes should consult [12] and [17].

The Zariski Topology. An affine variety V ⊆ ℂ^{n} has two topologies we will use. The first is the classical topology, induced from the usual topology on ℂ^{n}. The second is the Zariski topology, where the Zariski closed sets are subvarieties of V (meaning affine varieties of ℂ^{n} contained in V) and the Zariski open sets are their complements. Since subvarieties are closed in the classical topology (polynomials are continuous), Zariski open subsets are open in the classical topology.

Given a subset S ⊆ V, its closure \overline{S} in the Zariski topology is the smallest subvariety of V containing S. We call \overline{S} the Zariski closure of S. It is easy to give examples where this differs from the closure in the classical topology.

Affine Open Subsets and Localization. Some Zariski open subsets of an affine variety V are themselves affine varieties. Given f ∈ ℂ[V] \ {0}, let

V_{f} = {p ∈ V | f(p) ≠ 0} ⊆ V.

Then V_{f} is Zariski open in V and is also an affine variety, as we now explain.

Let V ⊆ ℂ^{n} have I(V) = ⟨f_{1}, …, f_{s}⟩ and pick g ∈ ℂ[x_{1}, …, x_{n}] representing f. Then V_{f} = V − V(g) is Zariski open in V. Now consider a new variable y and let W = V(f_{1}, …, f_{s}, 1 − gy) ⊆ ℂ^{n} × ℂ. Since the projection map ℂ^{n} ×ℂ → ℂ^{n} maps W bijectively onto V_{f}, we can identify V_{f} with the affine variety W ⊆ ℂ^{n} × ℂ.

When V is irreducible, the coordinate ring of V_{f} is easy to describe. Let ℂ(V) be the field of fractions of the integral domain ℂ[V]. Recall that elements of ℂ(V) give rational functions on V. Then let

(0.2)  ℂ[V]_{f} = {g/f^{ℓ} ∈ ℂ(V) | g ∈ ℂ[V], ℓ ≥ 0}.

In Exercise 0.2 you will prove that Spec(ℂ[V]_{f}) is the affine variety V_{f}.

Example 0.2. The n-dimensional torus is the affine open subset

(ℂ^{∗})^{n} = ℂ^{n} \ V(x_{1} ⋯ x_{n}) ⊆ ℂ^{n},

with coordinate ring

ℂ[x_{0}, …, x_{n}]_{x_{1}···x_{n}} = ℂ[x_{1}^{±1}, …, x_{n}^{±1}].

Elements of this ring are called Laurent polynomials. ⋄

The ring ℂ[V]_{f} from (0.2) is an example of localization. In Exercise 0.2 you will show how to construct this ring for all affine varieties, not just irreducible ones. The general concept of localization is discussed in standard texts in commutative algebra such as [2, Chapter 3] and [11, Chapter 2].

Normal Affine Varieties. Let R be an integral domain with field of fractions K. Then R is normal if every element of K which is integral over R (meaning that it is a root of a monic polynomial in R[x]) actually lies in R. For example, any UFD is normal (Exercise 0.3).

Definition 0.3. An irreducible affine variety V is normal if its coordinate ring ℂ[V] is normal.

For example, ℂ^{n} is normal since its coordinate ring ℂ[x_{1}, …, x_{n}] is a UFD and hence normal. Here is an example of a non-normal affine variety.

Example 0.4. Let C = V(x^{3} − y^{2}) ⊆ ℂ^{2}. This is an irreducible plane curve with a cusp at the origin. It is easy to see that ℂ[C] = ℂ[x, y]/⟨x^{3} − y^{2}⟩. Now let x̄ and ȳ be the cosets of x and y in ℂ[C] respectively. This gives ȳ/x̄ ∈ ℂ(C). A computation shows that ȳ/x̄ ∉ ℂ[C] and that (ȳ/x̄)^{2} = x̄. Consequently ℂ[C] and hence C are not normal.

We will see below that C is an affine toric variety. ⋄

An irreducible affine variety V has a normalization defined as follows. Let

ℂ[V]′ = {α ∈ ℂ(V) : α is integral over ℂ[V]}.

We call ℂ[V]′ the integral closure of ℂ[V]. One can show that ℂ[V]′ is normal and (with more work) finitely generated as a ℂ-algebra (see [11, Corollary 13.13]). This gives the normal affine variety

V′ = Spec(ℂ[V]′)

We call V′ the normalization of V. The natural inclusion ℂ[V] ⊆ ℂ[V]′ = ℂ[V′] corresponds to a map V′ → V. This is the normalization map.

Example 0.5. We saw in Example 0.4 that the curve C ⊆ ℂ^{2} defined by x^{3} = y^{2} has elements X, Y ∈ ℂ[C] such that Y/X ∉ ℂ[C] is integral over ℂ[C]. In Exercise 0.4 you will show that ℂ[ȳ/x̄] ⊆ ℂ(C) is the integral closure of ℂ[C] and that the normalization map is the map ℂ → C defined by t ↦ (t^{2}, t^{3}). ⋄

At first glance, the definition of normal does not seem very intuitive. Once we enter the world of toric varieties, however, we will see that normality has a very nice combinatorial interpretation and that the nicest toric varieties are the normal ones. We will also see that normality leads to a nice theory of divisors.

In Exercise 0.5 you will prove some properties of normal domains that will be used in §3 when we study normal affine toric varieties.

Smooth Points of Affine Varieties. In order to define a smooth point of an affine variety V, we first need to define local rings and Zariski tangent spaces. For simplicity, we assume that V is irreducible. The local ring of V at p is

O_{V,p} = {f/g ∈ ℂ(V) | f, g ∈ ℂ[V] and g(p) ≠ 0}.

Thus O_{V,p} consists of all rational functions on V that are defined at p. Inside of O_{V,p} we have the maximal ideal

𝔪_{V,p} = {φ ∈ O_{V,p} | φ(p) = 0}.

One can prove without difficulty that 𝔪_{V,p} is the unique maximal ideal of O_{V,p}. Rings with this property are called local rings.

The Zariski tangent space of V at p is defined to be

T_{p}(V) = Hom_{ℂ}(𝔪_{V,p}/𝔪_{V,p}^{2}, ℂ).

In Exercise 0.6 you will verify that dim T_{p}(ℂ^{n}) = n for every p ∈ ℂ^{n}. According to [17, p. 32], we can compute the Zariski tangent space of a point in an affine variety as follows.

Lemma 0.6. Let V ⊆ ℂ^{n} be a affine variety and let p ∈ V. Also assume that I(V) = ⟨f_{1}, …, f_{s}⟩ ⊆ ℂ[x_{1}, …, x_{n}]. For each i, let

d_{p}(f_{i}) = \frac{∂f_{i}}{∂x_{1}} (p) x_{1} + ⋯ + \frac{∂f_{i}}{∂x_{n}} (p) x_{n}.

The the Zariski tangent T_{p}(V) is isomorphic to the subspace of ℂ^{n} defined by the equations d_{p}(f_{1}) = ⋯ = d_{p}(f_{s}) = 0. In particular, dim T_{p}(V) ≤ n. □

Definition 0.7. A point p of an affine variety V is smooth or nonsingular if dim T_{p}(V) = dim V. The point p is singular if it is not smooth. Finally, V is smooth if every point of V is smooth.

Since dim T_{p}(ℂ^{n}) = n for every p ∈ ℂ^{n}, we see that ℂ^{n} is smooth. For an irreducible affine variety V ⊆ ℂ^{n} of dimension d, fix p ∈ V and write I(V) = ⟨f_{1}, …, f_{s}⟩. Using Lemma 0.6, it is straightforward to show that V is smooth at p if and only if the Jacobian matrix

(0.3)  J_{p}(f_{1}, …, f_{s}) = (\frac{∂f_{i}}{∂x_{j}} (p))_{1≤i≤s,1≤j≤n}

has rank n − d (Exercise 0.7). Here is a simple example.

Example 0.8. As noted in Example 0.4, the plane curve C defined by x^{3} = y^{2} has I(C) = ⟨x^{3} − y^{2}⟩ ⊆ ℂ[x, y]. A point p = (a, b) ∈ C has Jacobian

J_{p} = (3a^{2}, 2b),

so the origin is the only singular point of C.

Since T_{p}(V) = Hom_{ℂ}(𝔪_{V,p}/𝔪_{V,p}^{2}, ℂ), we see that V is smooth at p when dim V equals the dimension of 𝔪_{V,p}/𝔪_{V,p}^{2} as a vector space over O_{V,p}/𝔪_{V,p}. In terms of commutative algebra, this means that p ∈ V is smooth if and only if O_{V,p} is a regular local ring. See [2, p. 123] or [11, 10.3].

We can relate smoothness and normality as follows.

Proposition 0.9. A smooth irreducible affine variety V is normal.

Proof. As noted above, the smoothness of V implies that O_{V,p} is a regular local ring for all p ∈ V. A powerful result of commutative algebra states that a regular local ring is a UFD (see [11, Theorem 19.19]). Hence O_{V,p} is normal for all p. In §0 of Chapter 3 we will see that

ℂ[V] = ⋂_{p∈V} O_{V,p},

which by Exercise 0.5 implies that ℂ[V] is normal. □

The converse of Propostion 0.9 can fail. We will see in §3 that the affine variety V(xy − zw) ⊆ ℂ^{4} is normal, yet V(xy − zw) is singular at the origin.

Products of Affine Varieties. Given affine varieties V_{1} and V_{2}, there are several ways to show that the cartesian product V_{1} ×V_{2} is an affine variety. The most direct way is to proceed as follows. Let V_{1} ⊆ ℂ^{m} = Spec(ℂ[x_{1}, …, x_{m}]) and V_{2} ⊆ ℂ^{n} = Spec(ℂ[y_{1}, …, y_{n}]). Take I(V_{1}) = ⟨f_{1}, …, f_{s}⟩ and I(V_{2}) = ⟨g_{1}, …, g_{t}⟩. Since the f_{i} and g_{j} depend on separate sets of variables, it follows that

V_{1} × V_{2} = V(f_{1}, …, f_{s}, g_{1}, …, g_{t}) ⊆ ℂ^{m+n}

is an affine variety.

A fancier method is to use the mapping properties of the product. This will also give an instrinic description of its coordinate ring. Given V_{1} and V_{2} as above, V_{1} × V_{2} should be an affine variety with projections π_{i} : V_{1} × V_{2} → V_{i} such that whenever we have a diagram

[figure: W; ϕ_{1}; ϕ_{2}; ν; V_{1} × V_{2}; π_{1}; π_{2}; V_{1}; V_{2}]

where ϕ_{i} : W → V_{i} are morphisms from an affine variety W, there should be a unique morphism ν (the dotted arrow) that makes the diagram commute, i.e., π_{i} ∘ ν = ϕ_{i}. For the coordinate rings, this means that whenever we have a diagram

[figure: ℂ[V_{2}]; π_{2}^{∗}; ϕ_{2}^{∗}; ℂ[V_{1}]; π_{1}^{∗}; ϕ_{1}^{∗}; ℂ[V_{1} × V_{2}]; ν^{∗}; ℂ[W]]

with ℂ-algebra homomorphisms ϕ_{i}^{∗} : ℂ[V_{i}] → ℂ[W], there should be a unique ℂ-algebra homomorphism ν^{∗} (the dotted arrow) that makes the diagram commute. By the universal mapping property of the tensor product of ℂ-algebras, there is exactly one algebra that does this (up to isomorphism), namely ℂ[V_{1}] ⊗_{ℂ} ℂ[V_{2}]. This is a finitely generated ℂ-algebra with no nilpotents, so it is the coordinate ring of an affine variety by Lemma 0.1. We could take this as the definition of V_{1} × V_{2}.

Example 0.10. Let V be an affine variety. Since ℂ^{n} = Spec(ℂ[y_{1}, …, y_{n}]), the product V × ℂ^{n} has coordinate ring

ℂ[V] ⊗_{ℂ} ℂ[y_{1}, …, y_{n}] = ℂ[V][y_{1}, …, y_{n}].

If V is contained in ℂ^{m} with I(V) = ⟨f_{1}, …, f_{s}⟩ ⊆ ℂ[x_{1}, …, x_{m}], it follows that

I(V × ℂ^{n}) = ⟨f_{1}, …, f_{s}⟩ ⊆ ℂ[x_{1}, …, x_{m}, y_{1}, …, y_{n}].

For later purposes, we also note that the coordinate ring of V × (ℂ^{∗})^{n} is

ℂ[V]⊗_{ℂ}ℂ[y_{1}^{±1}, …, y_{n}^{±1}] = ℂ[V][y_{1}^{±1}, …, y_{n}^{±1}]. ⋄

Given affine varieties V_{1} and V_{2}, we note that the Zariski topology on V_{1} × V_{2} is usually not the product of the Zariski topologies on V_{1} and V_{2}.

Example 0.11. Consider ℂ^{2} = ℂ × ℂ. By definition, a basis for the product of the Zariski topologies consists of sets U_{1} × U_{2} where U_{i} are Zariski open in ℂ. Such a set is the complement of a union of collections of “horizontal” and “vertical” lines 1. Introduction to Affine Toric Varieties 9

in ℂ^{2}. This makes it easy to see that Zariski closed sets in ℂ^{2} such as V(y − x^{2}) cannot be closed in the product topology. ⋄

Exercises for § 0.

0.1. Prove Lemma 0.1. Hint: You will need the Nullstellensatz.

0.2. Let R be a finitely generated ℂ-algebra without nilpotents as in Lemma 0.1 and let f ∈ R be nonzero. The localization R_{f} consists of all formal expressions g/f^{k}, g ∈ R, modulo the equivalence relation

g/f^{k} ∼ h/f^{ℓ} ⟺ f^{s}(f^{ℓ}g − f^{k}h) = 0 for some s ≥ 0.

(a) Show that R_{f} satisfies Spec(R_{f}) = Spec(R)_{f}.

(b) Show that R_{f} is given by (0.2) when R is an integral domain.

0.3. Prove that a UFD is normal.

0.4. In the setting of Example 0.5, show that ℂ[ȳ/x̄] ⊆ ℂ(C) is the integral closure of ℂ[C] and that the normalization ℂ → C is defined by t ↦ (t^{2}, t^{3}).

0.5. In this exercise, you will prove some properties of normal domains need for §3.

(a) Let R be a normal domain with field of fractions K and fix f ∈ R. Prove that the localization R_{f} is normal.

(b) Let R_{α}. α ∈ A, be normal domains with the same field of fractions K. Prove that the intersection ⋂_{α∈A} R_{α} is normal.

0.6. Prove that dim T_{p}(ℂ^{n}) = n for all p ∈ ℂ^{n}.

0.7. Use Lemma 0.6 to prove the claim made in the text that smoothness is determined by the rank of the Jacobian matrix (0.3).

1. Introduction to Affine Toric Varieties

We first discuss what we mean by “torus” and then explore various constructions of affine toric varieties.

The Torus. The affine variety (ℂ^{∗})^{n} is a group under component-wise multiplication. A torus T is an affine variety isomorphic to (ℂ^{∗})^{n}, where T inherits a group structure from the isomorphism. Associated to T are its characters and oneparameter subgroups. We discuss each of these briefly.

A character of a torus T is a morphism χ : T → ℂ^{∗} that is a group homomorphism. For example, m = (a_{1}, …, a_{n}) ∈ ℤ^{n} gives a character χ^{m} : (ℂ^{∗})^{n} → ℂ^{∗} defined by

(1.1)  χ^{m}(t_{1}, …, t_{n}) = t_{1}^{a_{1}} ⋯ t_{n}^{a_{n}}.

One can show that all characters of (ℂ^{∗})^{n} arise this way (see [21, §16]). Thus the characters of (ℂ^{∗})^{n} form a group isomorphic to ℤ^{n}.

For an arbitrary torus T, its characters form a free abelian group M of rank equal to the dimension of T. It is customary to say that m ∈ M gives the character χ^{m} : T → ℂ^{∗}.

We will need the following results concerning tori (see [21, §16] for proofs).

Proposition 1.1.

(a) Let T_{1} and T_{2} be tori and let Φ : T_{1} → T_{2} be a morphism that is a group homomorphism. Then the image of Φ is a torus and is closed in T_{2}.

(b) Let T be a torus and let H ⊆ T be an irreducible subvariety of T that is a subgroup. Then H is a torus. □

Now assume that a torus T acts linearly on a finite dimensional vector space W over ℂ, where the action of t ∈ T on w ∈ W is denoted t · w. Given m ∈ M, we get the eigenspace

W_{m} = {w ∈ W | t · w = χ^{m}(t)w for all t ∈ T }.

If W_{m} ≠ {0}, then every w ∈ W_{m} \ {0} is a simultaneous eigenvector for all t ∈ T, with eigenvalue given by χ^{m}(t).

Proposition 1.2. In the above situation, we have W = ⨁_{m∈M} W_{m}. □

This proposition is a sophisticated way of saying that a family of commuting diagonalizable linear maps can be simultaneously diagonalized.

A one-parameter subgroup of a torus T is a morphism χ : ℂ^{∗} → T that is a group homomorphism. For example, u = (b_{1}, …, b_{n}) ∈ ℤ^{n} gives a one-parameter subgroup λ^{u} : ℂ^{∗} → (ℂ^{∗})^{n} defined by

(1.2)  λ^{u}(t) = (t^{b_{1}}, …, t^{b_{n}}).

All one-parameter subgroups of (ℂ^{∗})^{n} arise this way (see [21, §16]). It follows that the group of one-parameter subgroups of (ℂ^{∗})^{n} is naturally isomorphic to ℤ^{n}. For an arbitrary torus T, the one-parameter subgroups form a free abelian group N of rank equal to the dimension of T. As with the character group, an element u ∈ N gives the one-parameter subgroup λ^{u} : ℂ^{∗} → T.

There is a natural bilinear pairing ⟨ , ⟩ : M × N → ℤ defined as follows.

• (Intrinsic) Given a character χ^{m} and a one-parameter subgroup λ^{u}, the composition χ^{m} ∘ λ^{u} : ℂ^{∗} → ℂ^{∗} is character of ℂ^{∗}, which is given by t ↦ t^{ℓ} for some ℓ ∈ ℤ. Then ⟨m, u⟩ = ℓ.

• (Concrete) If T = (ℂ^{∗})^{n} with m = (a_{1}, …, a_{n}) ∈ ℤ^{n}, u = (b_{1}, …, b_{n}) ∈ ℤ^{n}, then one computes that

(1.3)  ⟨m, u⟩ = ∑_{i=1}^{n} a_{i}b_{i},

i.e., the pairing is the usual dot product.

It follows that the characters and one-parameter subgroups of a torus T form free abelian groups M and N of finite rank with a pairing ⟨ , ⟩ : M × N → ℤ that identifies N with Hom_{ℤ}(M, ℤ) and M with Hom_{ℤ}(N, ℤ). In terms of tensor products, one obtains a canonical isomorphism N ⊗_{ℤ} ℂ^{∗} ≃ T via u ⊗ t ↦ λ^{u}(t). Hence it is customary to write a torus as T_{N}.

From this point of view, picking an isomorphism T_{N} ≃ (ℂ^{∗})^{n} induces dual bases of M and N, i.e., isomorphisms M ≃ ℤ^{n} and N ≃ ℤ^{n} that turn characters into Laurent monomials (1.1), one-parameter subgroups into monomial curves (1.2), and the pairing into dot product (1.3).

The Definition of Affine Toric Variety. We now define the main object of study of this chapter.

Definition 1.3. An affine toric variety is an irreducible affine variety V containing a torus T_{N} ≃ (ℂ^{∗})^{n} as a Zariski open subset such that the action of T_{N} on itself extends to an action of T_{N} on V.

Obvious examples of affine toric varieties are (ℂ^{∗})^{n} and ℂ^{n}. Here are some less trivial examples.

Example 1.4. The plane curve C = V(x^{3} − y^{2}) ⊆ ℂ^{2} has a cusp at the origin. This is an affine toric variety with torus

C − {0} = C ∩ (ℂ^{∗})^{2} = {(t^{2}, t^{3}) | t ∈ ℂ^{∗}} ≃ ℂ^{∗},

where the isomorphism is (t^{2}, t^{3}) ↤ t. Example 0.4 shows that C is a non-normal toric variety. ⋄

Example 1.5. The variety V = V(xy − zw) ⊆ ℂ^{4} is a toric variety with torus

V ∩ (ℂ^{∗})^{4} = {(t_{1}, t_{2}, t_{3}, t_{1}t_{2}t_{3}^{−1}) | t_{i} ∈ ℂ^{∗}} ≃ (ℂ^{∗})^{3},

where the isomorphism is (t_{1}, t_{2}, t_{3}, t_{1}t_{2}t_{3}^{−1}) ↤ (t_{1}, t_{2}, t_{3}). We will see later that V is normal. ⋄

Example 1.6. Consider the surface in ℂ^{d+1} parametrized by the map

Φ : ℂ^{2} ⟶ ℂ^{d+1}

defined by (s, t) ↦ (s^{d}, s^{d−1}t, …, st^{d−1}, t^{d}). Thus Φ is defined using all degree d monomials in s, t.

Let the coordinates of ℂ^{d+1} be x_{0}, …, x_{d} and let I ⊆ ℂ[x_{0}, …, x_{d}] be the ideal generated by the 2 × 2 minors of the matrix

(\begin{matrix}x_{0} & x_{1} & ⋯ & x_{d−2} & x_{d−1} \\ x_{1} & x_{2} & ⋯ & x_{d−1} & x_{d} \end{matrix}) ,

so I = ⟨x_{i}x_{j+1} − x_{i+1}x_{j} | 0 ≤ i < j ≤ d − 1⟩. In Exercise 1.1 you will verify that Φ(ℂ^{2}) = V(I), so that Ĉ_{d} = Φ(ℂ^{2}) is an affine variety, and that I(Ĉ_{d}) = I, so that I is the ideal of all polynomials vanishing on Ĉ_{d}. It follows that I is prime since V is irreducible by Proposition 1.8 below. The affine surface Ĉ_{d} is called the rational normal cone of degree d and is an example of a determinantal variety. We will see below that I is an example of a toric ideal.

It is straightforward to show that Ĉ_{d} is a toric variety with torus

Φ((ℂ^{∗})^{2}) = Ĉ_{d} ∩ (ℂ^{∗})^{d+1} ≃ (ℂ^{∗})^{2}.

We will study this example from the projective point of view in Chapter 2. ⋄

We next explore three equivalent ways of constructing affine toric varieties.

Lattice Points. In this book, a lattice is a free abelian group of finite rank. Thus a lattice of rank n is isomorphic to ℤ^{n}. For example, a torus T_{N} has lattices M (of characters) and N (of one-parameter subgroups).

Given a torus T_{N} with character lattice M, a set A = {m_{1}, …, m_{s}} ⊆ M gives characters χ^{m_{i}} : T_{N} → ℂ^{∗}. Then consider the map

(1.4)  Φ_{A} : T_{N} ⟶ ℂ^{s}

defined by

Φ_{A} (t) = (χ^{m_{1}}(t), …, χ^{m_{s}}(t)) ∈ ℂ^{s}.

Definition 1.7. Given a finite set A ⊆ M, the affine toric variety Y_{A} is defined to be the Zariski closure of the image of the map Φ_{A} from (1.4).

This definition is justified by the following proposition.

Proposition 1.8. Given A ⊆ M as above, let ℤA ⊆ M be the sublattice generated by A. Then Y_{A} is an affine toric variety whose torus has character lattice ℤA. In particular, the dimension of Y_{A} is the rank of ℤA.

Proof. The map (1.4) can be regarded as a map

Φ_{A} : T_{N} ⟶ (ℂ^{∗})^{s}

of tori. By Proposition 1.1, the image T = Φ_{A} (T_{N}) is a torus that is closed in (ℂ^{∗})^{s}. The latter implies that Y_{A} ∩ (ℂ^{∗})^{s} = T since Y_{A} is the Zariski closure of the image. It follows that the image is Zariski open in Y_{A} . Furthermore, T is irreducible (it is a torus), so the same is true for its Zariski closure Y_{A} .

We next consider the action of T. Since T ⊆ (ℂ^{∗})^{s}, an element t ∈ T acts on ℂ^{s} and takes varieties to varieties. Then

T = t · T ⊆ t · Y_{A}

shows that t · Y_{A} is a variety containing T. Hence Y_{A} ⊆ t · Y_{A} by the definition of Zariski closure. Replacing t with t^{−1} leads to Y_{A} = t · Y_{A} , so that the action of T induces an action on Y_{A} . We conclude that Y_{A} is an affine toric variety.

It remains to compute the character lattice M(T) of T. Since T = Φ_{A} (T_{N}), the map Φ_{A} gives the commutative diagram

\xymatrix{T_{N} \ar[r]^{Φ_{A}} \ar@{->>}[dr] & (ℂ^{∗})^{s} \\  & T \ar@{^{(}->}[u]}

where ↠ denotes a surjective map and ↪ an injective map. This diagram of tori induces a commutative diagram of character lattices

\xymatrix{M & ℤ^{s} \ar[l]_{Φ̂_{A}} \ar@{->>}[d] \\  & M(T). \ar@{^{(}->}[ul]}

Since Φ̂_{A} : ℤ^{s} → M takes the standard basis e_{1}, …, e_{s} to m_{1}, …, m_{s}, the image of Φ̂_{A} is ℤA. By the diagram, we obtain M(T) ≃ ℤA. Then we are done since the dimension of a torus equals the rank of its character lattice. □

In concrete terms, fix a basis of M, so that we may assume M = ℤ^{n}. Then the s vectors in A ⊆ ℤ^{n} can be regarded as the columns of an n × s matrix A with integer entries. In this case, the the dimension of Y_{A} is simply the rank of the matrix A.

We will see below that every affine toric variety is isomorphic to Y_{A} for some finite subset A of a lattice.

Toric Ideals. Let Y_{A} ⊆ ℂ^{s} = Spec(ℂ[x_{1}, …, x_{s}]) be the affine toric variety coming from a finite set A = {m_{1}, …, m_{s}} ⊆ M. We can describe the ideal I(Y_{A} ) ⊆ ℂ[x_{1}, …, x_{s}] as follows. As in the proof of Proposition 1.8, (1.4)induces a map of character lattices

Φ̂_{A} : ℤ^{s} ⟶ M

that sends the standard basis e_{1}, …, e_{s} to m_{1}, …, m_{s}. Let L be the kernel of this map, so that we have an exact sequence

0 ⟶ L ⟶ ℤ^{s} ⟶ M.

In down to earth terms, elements ℓ = (ℓ_{1}, …, ℓ_{s}) of L satisfy ∑_{i=1}^{s} ℓ_{i}m_{i} = 0 and hence record the linear relations among the m_{i}.

Given ℓ = (ℓ_{1}, …, ℓ_{s}) ∈ L, set

ℓ_{+} = ∑_{ℓ_{i}>0} ℓ_{i}e_{i} and ℓ_{−} = − ∑_{ℓ_{i}<0} ℓ_{i}e_{i}.

Note that ℓ = ℓ_{+} − ℓ_{−} and that ℓ_{+}, ℓ_{−} ∈ ℕ^{s}. It follows easily that the binomial

x^{ℓ_{+}} − x^{ℓ_{−}} = ∏_{ℓ_{i}>0} x_{i}^{ℓ_{i}} − ∏_{ℓ_{i}<0} x_{i}^{−ℓ_{i}}

vanishes on the image of (1.4) and hence on Y_{A} since Y_{A} is the Zariski closure of the image.

Proposition 1.9. The ideal of the affine toric variety Y_{A} ⊆ ℂ^{s} is

I(Y_{A} ) = ⟨x^{ℓ_{+}} − x^{ℓ_{−}} | ℓ ∈ L⟩ = ⟨x^{α} − x^{β} | α, β ∈ ℕ^{s} and α − β ∈ L⟩.

Proof. We leave it to the reader to prove equality of the two ideals on the right (Exercise 1.2). Let I_{L} denote this ideal and note that I_{L} ⊆ I(Y_{A} ). We prove the opposite inclusion following [33, Lemma 4.1]. Pick a monomial order > on ℂ[x_{1}, …, x_{s}] and an isomorphism T_{N} ≃ (ℂ^{∗})^{n}. Thus we may assume M = ℤ^{n} and the map Φ : (ℂ^{∗})^{n} → ℂ^{s} is given by Laurent monomials t^{m_{i}} in variables t_{1}, …, t_{n}. If I_{L} ≠ I(Y_{A} ), then we can pick f ∈ I(Y_{A} ) \ I_{L} with minimal leading monomial x^{α} = ∏_{i=1}^{s} x_{i}^{a_{i}}. Rescaling if necessary, x^{α} becomes the leading term of f.

Since f(t^{m_{1}}, …, t^{m_{s}}) is identically zero as a polynomial in t_{1}, …, t_{n}, there must be cancellation involving the term coming from x^{α}. In other words, f must contain a monomial x^{β} = ∏_{i=1}^{s} x_{i}^{b_{i}} < x^{α} such that

∏_{i=1}^{s}(t^{m_{i}})^{a_{i}} = ∏_{i=1}^{s}(t^{m_{i}})^{b_{i}}.

This implies that

∑_{i=1}^{s} a_{i}m_{i} = ∑_{i=1}^{s} b_{i}m_{i},

so that α−β = ∑_{i=1}^{s}(a_{i}−b_{i})e_{i} ∈ L. Then x^{α}−x^{β} ∈ I_{L} by the second description of I_{L}. It follows that f − x^{α} + x^{β} also lies in I(Y_{A} ) \ I_{L} and has strictly smaller leading term. This contradiction completes the proof. □

Given a finite set A ⊆ M, there are several methods to compute the ideal I(Y_{A} ) = I_{L} of Proposition 1.9. For simple examples, the rational implicitization algorithm of [7, §3 of Chapter 3] can be used. It is also possible to compute I_{L} using a basis of L and ideal quotients (Exercise 1.3). Further comments on computing I_{L} can be found in [33, Chapter 12].

Inspired by Proposition 1.9, we make the following definition.

Definition 1.10. Let L ⊆ ℤ^{s} be a sublattice.

(a) The ideal I_{L} = ⟨x^{α} − x^{β} | α, β ∈ ℕ^{s} and α − β ∈ L⟩ is called a lattice ideal.

(b) A prime lattice ideal is called a toric ideal.

Since toric varieties are irreducible, the ideals appearing in Proposition 1.9 are toric ideals. Examples of toric ideals include:

Example 1.4 : ⟨x^{3} − y^{2}⟩ ⊆ ℂ[x, y]
Example 1.5 : ⟨xz − yw⟩ ⊆ ℂ[x, y, z, w]
Example 1.6 : ⟨x_{i}x_{j+1} − x_{i+1}x_{j} | 0 ≤ i < j ≤ d − 1⟩ ⊆ ℂ[x_{0}, …, x_{d}].

(The latter is the ideal of the rational normal cone Ĉ_{d} ⊆ ℂ^{d+1}.) In each example, we have a prime ideal generated by binomials. As we now show, such ideal are automatically toric.

Proposition 1.11. An ideal I ⊆ ℂ[x_{1}, …, x_{s}] is toric if and only if it is prime and generated by binomials.

Proof. One direction is obvious. So suppose that I is prime and generated by binomials x^{α_{i}} − x^{β_{i}}. Then observe that V(I) ∩ (ℂ^{∗})^{s} is nonempty (it contains (1, …, 1)) and is a subgroup of (ℂ^{∗})^{s} (easy to check). Since V(I) ⊆ ℂ^{s} is irreducible, it follows that V(I) ∩ (ℂ^{∗})^{s} is an irreducible subvariety of (ℂ^{∗})^{s} that is also a subgroup. By Proposition 1.1, we see that T = V(I) ∩ (ℂ^{∗})^{s} is a torus.

Projecting on the ith coordinate of (ℂ^{∗})^{s} gives a character T ↪ (ℂ^{∗})^{s} → ℂ^{∗}, which by our usual convention we write as χ^{m_{i}} : T → ℂ^{∗} for m_{i} ∈ M(T). It follows easily that V(I) = Y_{A} for A = {m_{1}, …, m_{s}}, and since I is prime, we have I = I(Y_{A} ) by the Nullstellensatz. Then I is toric by Proposition 1.9. □

We will later see that all affine torics arise from toric ideals. For more on toric ideals and lattice ideals, the reader should consult [23, Chapter 7].

Affine Semigroups. A semigroup is a set S with an associative binary operation and an identity element. To be an affine semigroup, we further require that

• The binary operation on S is commutative. We will write the operation as + and the identity element as 0. Thus a finite set A ⊆ S gives

ℕA = {_{∑m∈A} a_{m}m | a_{m} ∈ ℕ} ⊆ S.

• The semigroup is finitely generated, meaning that there is a finite set A ⊆ S such that ℕA = S.

• The semigroup can be embedded in a lattice M. The simplest example of an affine semigroup is ℕ^{n} ⊆ ℤ^{n}. More generally, given a lattice M and a finite set A ⊆ M, we get the affine semigroup ℕA ⊆ M. Up to isomorphism, all affine semigroups are of this form.

Given an affine semigroup S ⊆ M, the semigroup algebra ℂ[S] is the vector space over ℂ with S as basis and multiplication induced by the semigroup structure of S. To make this precise, we think of M as the character lattice of a torus T_{N}, so that m ∈ M gives the character χ^{m}. Then

ℂ[S] = { ∑_{m∈S} c_{m}χ^{m} | c_{m} ∈ ℂ and c_{m} = 0 for all but finitely manym},

with multiplication induced by

χ^{m} · χ^{m′} = χ^{m+m′}.

If S = ℕA for A = {m_{1}, …, m_{s}}, then ℂ[S] = ℂ[χ^{m_{1}}, …, χ^{m_{s}}].

Here are two basic examples.

Example 1.12. The affine semigroup ℕ^{n} ⊆ ℤ^{n} gives the polynomial ring

ℂ[ℕ^{n}] = ℂ[x_{1}, …, x_{n}],

where x_{i} = χ^{e_{i}} and e_{1}, …, e_{n} is the standard basis of ℤ^{n}. ⋄

Example 1.13. If e_{1}, …, e_{n} is a basis of a lattice M, then M is generated by A = {±e_{1}, …, ±e_{n}} as an affine semigroup. Setting t_{i} = χ^{e_{i}} gives the Laurent polynomial ring

ℂ[M] = ℂ[t_{1}^{±1}, …, t_{n}^{±1}].

Using Example 0.2, one sees that ℂ[M] is the coordinate ring of the torus T_{N}. ⋄

Affine semigroup rings give rise to affine toric varieties as follows.

Proposition 1.14. Let S ⊆ M be an affine semigroup.

(a) ℂ[S] is an integral domain and finitely generated as a ℂ-algebra.

(b) Spec(ℂ[S]) is an affine toric variety whose torus has character lattice ℤS, and if S = ℕA for a finite set A ⊆ M, then Spec(ℂ[S]) = Y_{A} .

Proof. As noted above, A = {m_{1}, …, m_{s}} implies ℂ[S] = ℂ[χ^{m_{1}}, …, χ^{m_{s}}], so ℂ[S] is finitely generated. Since ℂ[S] ⊆ ℂ[M] follows from S ⊆ M, we see that ℂ[S] is an integral domain by Example 1.13.

Using A = {m_{1}, …, m_{s}}, we get the ℂ-algebra homomorphism

π : ℂ[x_{1}, …, x_{s}] ⟶ ℂ[M]

where x_{i} ↦ χ^{m_{i}} ∈ M. This corresponds to the morphism

Φ_{A} : T_{N} ⟶ ℂ^{s}

from (1.4), i.e., π = (Φ_{A} )^{∗} in the notation of §0. One checks that the kernel of π is the toric ideal I(Y_{A} ) (Exercise 1.4). Since the image of π is ℂ[χ^{m_{1}}, …, χ^{m_{s}}] = ℂ[S], the coordinate ring of Y_{A} is

(1.5)  \begin{aligned}ℂ[Y_{A} ] & = ℂ[x_{1}, …, x_{n}]/I(Y_{A} ) \\ & = ℂ[x_{1}, …, x_{n}]/Ker(π) ≃ Im(π) = ℂ[S]. \end{aligned}

Thus Spec(ℂ[S]) is the toric variety Y_{A} . Since S = ℕA implies ℤS = ℤA, the torus of Y_{A} = Spec(ℂ[S]) has the desired character lattice by Proposition 1.8. □

Here is an example of this proposition.

Example 1.15. Consider the affine semigroup S ⊆ ℤ generated by 2 and 3, so that S = {0, 2, 3, … }. To study the semigroup algebra ℂ[S], we use (1.5). If A = {2, 3}, then Φ_{A} (t) = (t^{2}, t^{3}) and the toric ideal is I(Y_{A} ) = ⟨x^{3} − y^{2}⟩ by Example 1.4. Hence

ℂ[S] = ℂ[t^{2}, t^{3}] ≃ ℂ[x, y]/⟨x^{3} − y^{2}⟩

and the affine toric variety Y_{A} is the curve x^{3} = y^{2}. ⋄

Equivalence of Constructions. We can now state the main result of this section, which asserts that our various approaches to affine toric varieties all give the same class of objects.

Theorem 1.16. Let V be an affine variety. The following are equivalent:

(a) V is an affine toric variety according to Definition 1.3.

(b) V = Y_{A} for a finite set A in a lattice.

(c) V is an affine variety defined by a toric ideal.

(d) V = Spec(ℂ[S]) for an affine semigroup S.

Proof. The implications (b) ⇔ (c) ⇔ (d) ⇒ (a) follow from Propositions 1.8, 1.9 and 1.14. For (a) ⇒ (d), let V be an affine toric variety containing the torus T_{N}, and let M be the character lattice of T_{N}. Since the coordinate ring of T_{N} is the semigroup algebra ℂ[M], the inclusion T_{N} ⊆ V induces a map of coordinate rings

ℂ[V] ⟶ ℂ[M].

This map is injective since T_{N} is Zariski dense in V, so that we can regard ℂ[V] as a subalgebra of ℂ[M]. Let

S = {m ∈ M | χ^{m} ∈ ℂ[V]}

and note that S is a semigroup. We will show that S is finitely generated with semigroup algebra equal to ℂ[V]. This will complete the proof of the theorem.

The inclusion ℂ[S] ⊆ ℂ[V] is obvious. For the opposite inclusion, pick f ≠ 0 in ℂ[V]. Using ℂ[V] ⊆ ℂ[M], we can write

f = ∑_{m∈B} c_{m}χ^{m},

where B ⊆ M is finite and c_{m} ≠ 0 for all m ∈ B. Let

W = Span(χ^{m} | m ∈ B) ⊆ M_{ℝ}

be the subspace spanned by the χ^{m}. Thus f ∈ W ∩ ℂ[V]. If t ∈ T_{N}, then the action of t on V induces an action on the coordinate ring ℂ[V], so that we get t · f ∈ ℂ[V]. By Definition 1.3, this extends the usual action of t on T_{N}, which in terms of the coordinate ring ℂ[M] is given by t·χ^{m} = χ^{m}(t)χ^{m}. It follows that W and hence W ∩ ℂ[V] are stable under the action of T_{N}. Since W ∩ ℂ[V] is finitedimensional, Proposition 1.2 implies that W ∩ ℂ[V] is spanned by simultaneous eigenvectors of T_{N}. But this is taking place in ℂ[M], where the simultaneous eigenvectors are the characters! It follows that W ∩ ℂ[V] is spanned by characters. Then the above expression for f ∈ W ∩ ℂ[V] implies that χ_{m} ∈ ℂ[V] for m ∈ B. It follows that f ∈ ℂ[S], proving that ℂ[V] = ℂ[S].

It remains to show that S is finitely generated. Since ℂ[V] is finitely generated, we can find f_{1}, …, f_{s} ∈ ℂ[V] with ℂ[V] = ℂ[f_{1}, …, f_{s}]. Expressing each f_{i} in terms of characters as above gives the desired finite generating set of S. Hence S is an affine semigroup. □

Here is one way to think about the above proof. When an irreducible affine variety V contains a torus T_{N} as a Zariski open subset, we have the inclusion

ℂ[V] ⊆ ℂ[M].

Thus ℂ[V] consists of those functions on the torus T_{N} that extend to polynomial functions on V. Then the key insight is that V is a toric variety precisely when the functions that extend are determined by the characters that extend.

Example 1.17. We’ve seen that V = V(xy−zw) ⊆ ℂ^{4} is a toric variety with toric ideal ⟨xy −zw⟩ ⊆ ℂ[x, y, z, w]. Also, the torus is (ℂ^{∗})^{3} via the map (t_{1}, t_{2}, t_{3}) ↦ (t_{1}, t_{2}, t_{3}, t_{1}t_{2}t_{3}^{−1}). The lattice points used in this map can be represented as the columns of the matrix

(1.6)  (\begin{matrix}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & −1 \end{matrix}) .

The corresponding semigroup S ⊆ ℤ^{3} consists of the ℕ-linear combinations of the column vectors. Hence the elements of S are lattice points lying in the polyhedral region in ℝ^{3} pictured in Figure 1. In this figure, the four vectors generating S are shown in bold, and the boundary of the polyhedral region is partially shaded. In the terminology of §2, this polyhedral region is a rational polyhedral cone. In Exercise 1.5 you will show that S consists of all lattice points lying in the cone in Figure 1. We will use this in §3 to prove that V is normal. ⋄

Exercises for § 1.

1.1. As in Example 1.6, let I = ⟨x_{i}x_{j+1} − x_{i+1}x_{j} | 0 ≤ i < j ≤ d − 1⟩ ⊆ ℂ[x_{0}, …, x_{d}]

and let Ĉ_{d} be the surface parametrized by

Φ(s, t) = (s^{d}, s^{d−1}t, …, st^{d−1}, t^{d}) ∈ ℂ^{d+1}.

1. Introduction to Affine Toric Varieties 19

Figure 1. Cone containing the lattice points corresponding to V = V(xy − zw)

(a) Prove that V(I) = Φ(ℂ^{2}) ⊆ ℂ^{d+1}. Thus Ĉ_{d} = V(I).

(b) Prove that I(Ĉ_{d}) is homogeneous.

(c) Consider lex monomial order with x_{0} > x_{1} > ⋯ > x_{d}. Let f ∈ I(Ĉ_{d}) be homogeneous of degree ℓ and let r be the remainder of f on division by the generators of I. Prove that r can be written

r = h_{0}(x_{0}, x_{1}) + h_{1}(x_{1}, x_{2}) + ⋯ + h_{d−1}(x_{d−1}, x_{d})

where h_{i} is homogeneous of degree ℓ. Also explain why we may assume that the coefficient of x_{i}^{ℓ} in h_{i} is zero for 1 ≤ i ≤ d − 1.

(d) Use part (c) and r(s^{d}, s^{d−1}t, …, st^{d−1}, t^{d}) = 0 to show that r = 0.

(e) Use parts (b), (c) and (d) to prove that I = I(Ĉ_{d}). Also explain why the generators of I are a Gröbner basis for the above lex order.

1.2. Let L ⊆ ℤ^{s} be a sublattice. Prove that

⟨x^{ℓ_{+}} − x^{ℓ_{−}} | ℓ ∈ L⟩ = ⟨x^{α} − x^{β} | α, β ∈ ℕ^{s}, α − β ∈ L⟩.

Note that when ℓ ∈ L, the vectors ℓ_{+}, ℓ_{−} ∈ ℕ^{s} have disjoint support (i.e., no coordinate is positive in both), while this may fail for arbitrary α, β ∈ ℕ^{s} with α − β ∈ L.

1.3. Let I_{L} be a toric ideal and let ℓ^{1}, …, ℓ^{r} be a basis of the sublattice L ⊆ ℤ^{s}. Define

I_{L} = ⟨x^{ℓ_{+}^{i}} − x^{ℓ_{−}^{i}} | i = 1, …, r⟩.

Prove that I_{L} = I_{L} : ⟨x_{1} ⋯ x_{s}⟩^{∞}. Hint: Given α, β ∈ ℕ^{s} with α − β ∈ L, write α − β = ∑_{i=1}^{r} a_{i}ℓ^{i}, a_{i} ∈ ℤ. This implies

x^{α−β} − 1 = ∏_{a_{i}>0} (\frac{x^{ℓ_{+}^{i}}}{x^{ℓ_{−}^{i}}} )^{a_{i}} ∏_{a_{i}<0} (\frac{x^{ℓ_{−}^{i}}}{x^{ℓ_{+}^{i}}} )^{−a_{i}} − 1.

20 1. Affine Toric Varieties

Show that multiplying this by (x_{1} ⋯ x_{s})^{k} gives an element of I_{L} for k ≫ 0. (By being more careful, one can show that this result holds for lattice ideals. See [23, Lemma 7.6].)

1.4. Fix an affine variety V and f_{1}, …, f_{s} ∈ ℂ[V]. This gives a polynomial map Φ : V → ℂ^{s}, which on coordinate rings is given by

Φ^{∗} : ℂ[x_{1}, …, x_{s}] ⟶ ℂ[V], x_{i} ⟼ f_{i}.

Let Y ⊆ ℂ^{s} be the Zariski closure of the image of Φ.

(a) Prove that I(Y) = Ker(Φ^{∗}).

(b) Explain how this applies to the proof of Proposition 1.14.

1.5. Let m_{1} = (1, 0, 0), m_{2} = (0, 1, 0), m_{3} = (0, 0, 1), m_{4} = (1, 1, −1) be the columns of the matrix in Example 1.17 and let

C = { ∑_{i=1}^{4} λ_{i}m_{i} | λ_{i} ∈ ℝ_{≥0}} ⊆ ℝ^{3}

be the cone in Figure 1. Prove that C ∩ ℤ^{3} is a semigroup generated by m_{1}, m_{2}, m_{3}, m_{4}.

1.6. An interesting observation is that different sets of lattice points can parametrize the same affine toric variety, even though these parametrizations behave slightly differently. In this exercise you will consider the parametrizations

Φ_{1}(s, t) = (s^{2}, st, st^{3}) and Φ_{2}(s, t) = (s^{3}, st, t^{3}).

(a) Prove that Φ_{1} and Φ_{2} both give the affine toric variety Y = V(xz − y^{3}) ⊆ ℂ^{3}.

(b) We can regard Φ_{1} and Φ_{2} as maps

Φ_{1} : ℂ^{2} ⟶ Y and Φ_{2} : ℂ^{2} ⟶ Y.

Prove that Φ_{2} is surjective and that Φ_{1} is not. In general, a finite subset A ⊆ ℤ^{n} gives a rational map Φ_{A} : ℂ^{n} −−→ Y_{A}. The image of Φ_{A} in ℂ^{s} is called a toric set in the literature. Thus Φ_{1}(ℂ^{2}) and Φ_{2}(ℂ^{2}) are toric sets. The papers [22] and [28] study when a toric set equals the corresponding affine toric variety.

1.7. In Example 1.6 and Exercise 1.1 we constructed the rational normal cone Ĉ_{d} using all monomials of degree d in s, t. If we drop some of the monomials, things become more complicated. For example, consider the surface parametrized by

Φ(s, t) = (s^{4}, s^{3}t, st^{3}, t^{4}) ∈ ℂ^{4}.

This gives a toric variety Y ⊆ ℂ^{4}. Show that the toric ideal of Y is given by

I(Y) = ⟨xw − yz, yw^{2} − z^{3}, xz^{2} − y^{2}w, x^{2}z − y^{3}⟩ ⊆ ℂ[x, y, z, w].

The toric ideal for Ĉ_{4} has quadratic generators; by removing the monomial s^{2}t^{2}, we now get cubic generators. In Chapter 2 we will use this example to construct a projective curve that is normal but not projectively normal.

1.8. Prove that a lattice ideal I_{L} for L ⊆ ℤ^{s} is a toric ideal if and only if ℤ^{s}/L is torsionfree. Hint: When ℤ^{s}/L is torsion-free, it can be regarded as the character lattice of a torus. The other direction of the proof is more challenging. If you get stuck, see [23, Theorem 7.4].

2. Cones and Affine Toric Varieties 21

1.9. Prove that I = ⟨x^{2} − 1, xy − 1, yz − 1⟩ is the lattice ideal for the lattice

L = {(a, b, c) ∈ ℤ^{3} | a + b + c ≡ 0 mod 2} ⊆ ℤ^{3}.

Also compute primary decomposition of I to show that I is not prime.

1.10. Let T_{N} be a torus with character lattice M. Then every point t ∈ T_{N} gives an evaluation map ϕ_{t} : M → ℂ^{∗} defined by ϕ_{t}(m) = χ^{m}(t). Prove that ϕ_{t} is a group homomorphism and that the map t ↦ ϕ_{t} induces a group isomorphism

T_{N} ≃ Hom_{ℤ}(M, ℂ^{∗}).

1.11. Consider tori T_{1} and T_{2} with character lattices M_{1} and M_{2}. By Example 1.13, the coordinate rings of T_{1} and T_{2} are ℂ[M_{1}] and ℂ[M_{2}]. Let Φ : T_{1} → T_{2} be a morphism that is a group homomorphism. Then Φ induces maps

Φ̂ : M_{2} ⟶ M_{1} and Φ^{∗} : ℂ[M_{2}] ⟶ ℂ[M_{1}]

by composition. Prove that Φ^{∗} is the map of semigroup algebras induced by the map Φ̂ of affine semigroups.

1.12. A commutative semigroup S is cancellative if u + v = u + w implies v = w for all u, v, w ∈ S and torsion-free if nu = 0 implies u = 0 for all n ∈ ℕ \ {0} and u ∈ S. Prove that S is affine if and only if it is finitely generated, cancellative, and torsion-free.

1.13. The requirement that an affine semigroup be finitely generated is important since lattices contain semigroups that are not finitely generated. For example, let τ > 0 be irrational and consider the semigroup

S = {(a, b) ∈ ℕ^{2} | b ≥ τa} ⊆ ℤ^{2}.

Prove that S is not finitely generated. (When τ satisfies a quadratic equation with integer coefficients, the generators of S are related to continued fractions. For example, when τ = (1+\sqrt{5})/2 is the golden ratio, the minimal generators of S are (1, 0) and (F_{2n−1}, F_{2n}) for n = 1, 2, …, where F_{n} is the nth Fibonacci number. See [32] for further details.)

1.14. Suppose that ϕ : M → M is a group isomorphism. Fix a finite set A ⊆ M and let B = ϕ(A). Prove that the toric varieties Y_{A} and Y_{B} are equivariantly isomorphic (meaning that the isomorphism respects the torus action).

2. Cones and Affine Toric Varieties

We begin with a brief discussion of rational polyhedral cones and then explain how they relate to affine toric varieties.

Polyhedral Cones. Fix a pair of dual vector spaces M_{ℝ} and N_{ℝ} of dimension n. Our discussion will omit most proofs—we refer the reader to [13] for a more detailed treatment and [26, Appendix A.1] for careful statements. See also [4, 15, 29].

Definition 2.1. A polyhedral cone in N_{ℝ} is a set of the form

σ = Cone(S) = { ∑_{u∈S} λ_{u}u | λ_{u} ≥ 0} ⊆ N_{ℝ},

where S ⊆ N_{ℝ} is finite. We say that σ is generated by S. Also set Cone(∅) = {0}.

One easily checks that polyhedral cone σ is convex, meaning that x, y ∈ σ ⇒ λx + (1 − λ)y ∈ σ for all 0 ≤ λ ≤ 1, and is a cone, meaning x ∈ σ ⇒ λx ∈ σ for all λ ≥ 0. Cones satisfying Definition 2.1 are sometimes called “convex polyhedral cones”, but we will drop the “convex” since all cones we consider will be convex.

Examples of convex polyhedral cones include the first quadrant in ℝ^{2} or first octant in ℝ^{3}. For another example, the cone Cone(e_{1}, e_{2}, e_{1} + e_{3}.e_{2} + e_{3}) ⊆ ℝ^{3} is pictured in Figure 2. It is also possible to have cones that contain entire lines.

Figure 2. Cone in ℝ^{3} generated by e_{1}, e_{2}, e_{1} + e_{3}.e_{2} + e_{3}

For example, Cone(e_{1}, −e_{1}) ⊆ ℝ^{2} is the x-axis, while Cone(e_{1}, −e_{1}, e_{2}) is the closed upper half-plane {(x, y) ∈ ℝ^{2} | y ≥ 0}. As we will see below, these last two examples are not strongly convex.

We can also create cones using polytopes, which are defined as follows.

Definition 2.2. A polytope in N_{ℝ} is a set of the form

P = Conv(S) = { ∑_{u∈S} λ_{u}u | λ_{u} ≥ 0, ∑_{u∈S} λ_{u} = 1} ⊆ N_{ℝ},

where S ⊆ N_{ℝ} is finite. We say that P is the convex hull of S.

Polytopes include all polygons in ℝ^{2} and polyhedra in ℝ^{3}. As we will see in later chapters, polytopes play a prominent role in the theory of toric varieties. Here, however, we simply observe that a polytope P ⊆ N_{ℝ} gives a polyhedral cone in N_{ℝ} × ℝ by taking the cone

σ = {λ · (u, 1) ∈ N_{ℝ} × ℝ | u ∈ P, λ ≥ 0}.

If P = Conv(S), then we can also describe this as σ = Cone(S × {1}). Figure 3 on the next page shows what this looks when P is a pentagon in the plane

Figure 3. Cone over a pentagon P ⊆ ℝ^{2}

The dimension of a polyhedral cone σ, denoted dim σ, is the dimension of the smallest subspace W of N_{ℝ} containing σ. We call W the span of σ.

Dual Cones and Faces. As usual, the pairing between M_{ℝ} and N_{ℝ} is denoted ⟨ , ⟩.

Definition 2.3. Given a polyhedral cone σ ⊆ N_{ℝ}, its dual cone is defined by

σ^{∨} = {m ∈ M_{ℝ} | ⟨m, u⟩ ≥ 0 for all u ∈ σ}.

Duality has the following important properties.

Proposition 2.4. Let σ ⊆ N_{ℝ} be a polyhedral cone. Then σ^{∨} is a polyhedral cone in M_{ℝ} and (σ^{∨})^{∨} = σ. □

Given m ≠ 0 in N_{ℝ}, we get the hyperplane

H_{m} = {u ∈ M_{ℝ} | ⟨m, u⟩ = 0}

and the closed half-space

H_{m}^{+} = {u ∈ M_{ℝ} | ⟨m, u⟩ ≥ 0}.

We say that H_{m} is a supporting hyperplane of a polyhedral cone σ if σ ⊆ H_{m}^{+}, and H_{m}^{+} is a supporting half-space. Note that H_{m} is a supporting hyperplane of σ if and only if m ∈ σ^{∨} \ {0}. Furthermore, if m_{1}, …, m_{s} generate σ^{∨}, then it is straightforward to check that

(2.1)  σ = H_{m_{1}}^{+} ∩ ⋯ ∩ H_{m_{1}}^{+}.

Thus every polyhedral cone is an intersection of finitely many closed half-spaces. We can use supporting hyperplanes and half-spaces to define faces of a cone.

Definition 2.5. A face of the polyhedral cone σ is τ = H_{m} ∩ σ for some m ∈ σ^{∨}. Using m = 0 shows that σ as a face of itself. Faces τ ≠ σ are called proper faces.

The faces of a polyhedral cone have the following obvious properties.

Lemma 2.6. Let σ = Cone(S) be a polyhedral cone. Then:

(a) Every face of σ is a polyhedral cone.

(b) An intersection of two faces of σ is again a face of σ.

(c) A face of a face of σ is again a face of σ. □

A facet of σ is a face τ of codimension 1, i.e., dim τ = dim σ − 1. An edge of σ is a face of dimension 1. In Figure 4 we illustrate a 3-dimensional cone with shaded facets and a supporting hyperplane (a plane in this case) that cuts out the vertical edge of the cone.

Figure 4. A cone σ ⊆ ℝ^{3} with shaded facets and a supporting hyperplane

Here are some properties of facets.

Proposition 2.7. Let σ ⊆ N_{ℝ} ≃ ℝ^{n} be a polyhedral cone. Then:

(a) If dim σ = n and the facets of σ are τ_{i} = H_{m_{i}} ∩ σ for m_{i} ∈ σ^{∨}, 1 ≤ i ≤ s, then

σ = H_{m_{1}}^{+} ∩ ⋯ ∩ H_{m_{s}}^{+} and σ^{∨} = Cone(m_{1}, …, m_{s}).

(b) Every proper face τ of σ is the intersection of the facets of σ containing τ. □

Note how part (a) of the proposition refines (2.1) when dim σ = dim N_{ℝ}. When working in ℝ^{n}, dot product allows us to identify the dual with ℝ^{n}. From

this point of view, the vectors m_{1}, …, m_{s} in part (a) of the proposition are facet normals, i.e., perpendicular to the facets. This makes it easy to compute examples.

Example 2.8. It easy to see that the facet normals to the cone σ ⊆ ℝ^{3} in Figure 2 are m_{1} = e_{1}, m_{2} = e_{2}, m_{3} = e_{3}, m_{4} = e_{1} + e_{2} − e_{3}. Hence

σ^{∨} = Cone(e_{1}, e_{2}, e_{3}, e_{1} + e_{2} − e_{3}) ⊆ ℝ^{3}.

This is the cone of Figure 1 at the end of §1 whose lattice points describe the semigroup of the affine toric variety V(xy − zw) (see Example 1.17). As we will see, this is part of how cones describe normal affine toric varieties.

Now consider σ^{∨}, which is the cone in Figure 1. The reader can check that the facet normals of this cone are e_{1}, e_{2}, e_{1} + e_{3}, e_{2} + e_{3}. Using duality and part (b) of Proposition 2.7, we obtain

σ = (σ^{∨})^{∨} = Cone(e_{1}, e_{2}, e_{1} + e_{3}, e_{2} + e_{3}).

Hence we recover our original description of σ. ⋄

In this example, facets of the cone correspond to edges of its dual. More generally, given a face τ of a polyhedral cone σ ⊆ N_{ℝ}, we define

τ^{⊥} = {m ∈ M_{ℝ} | ⟨m, u⟩ = 0 for all u ∈ τ}
τ^{∗} = {m ∈ σ^{∨} | ⟨m, u⟩ = 0 for all u ∈ τ}
= σ^{∨} ∩ τ^{⊥}.

We call τ^{∗} the dual face of τ because of the following proposition.

Proposition 2.9. If τ is a face of a polyhedral cone σ and τ^{∗} = σ^{∨} ∩ τ^{⊥}, then:

(a) τ^{∗} is a face of σ^{∨}.

(b) The map τ ↦ τ^{∗} is a bijective inclusion-reversing correspondance between the faces of σ and the faces of σ^{∨}.

(c) dim τ + dim τ^{∗} = n. □

Here is an example of Proposition 2.9 when dim σ < dim N_{ℝ}.

Example 2.10. Let σ = Cone(e_{1}, e_{2}) ⊆ ℝ^{3}. Figure 5 shows σ and σ^{∨}. You

Figure 5. A 2-dimensional cone σ ⊆ ℝ^{3} and its dual σ^{∨} ⊆ ℝ^{3}

should check that the maximal face of σ, namely σ itself, gives the minimal face σ^{∗} of σ^{∨}, namely the z-axis. Note also that

dim σ + dim σ^{∗} = 3

even though σ has dimension 2. ⋄

Strong Convexity. Of the cones shown in Figures 1–5, all but σ^{∨} in Figure 5 have the nice property that the origin is a face. Such cones are called strongly convex. This condition can be stated several ways.

Proposition 2.11. Let σ ⊆ N_{ℝ} ≃ ℝ^{n} be a polyhedral cone. Then:

σ is strongly convex ⟺ {0} is a face of σ

⟺ σ contains no positive-dimensional subspace ⟺ σ ∩ (−σ) = {0}

⟺ dim σ^{∨} = n. □

You will prove Proposition 2.11 in Exercise 2.1. One corollary is that if a polyhedral cone σ is strongly convex of maximal dimension, then so is σ^{∨}. The cones pictured in Figures 1–4 satisfy this condition.

In general, a polyhedral cone σ always has a minimal face that is the largest subspace W contained in σ. Furthermore, we have W = σ ∩ (−σ) = (σ^{∨})^{⊥}, and \overline{σ} = σ/W ⊆ N_{ℝ}/W is a strongly convex polyhedral cone. See Exercise 2.2.

Rational Polyhedral Cones. Let N and M be dual lattices with associated vector spaces N_{ℝ} = N ⊗_{ℤ} ℝ and M_{ℝ} = M ⊗_{ℤ} ℝ. For ℝ^{n} we usually use the lattice ℤ^{n}.

Definition 2.12. A polyhedral cone σ ⊆ N_{ℝ} is rational if σ = Cone(S) for some finite set S ⊆ N.

The cones appearing in Figures 1, 2 and 5 are rational. We note without proof that faces and duals of rational polyhedral cones are rational. Furthermore, if σ = Cone(S) for S ⊆ N finite and N_{ℚ} = N ⊗_{ℤ} ℚ, then

(2.2)  σ ∩ N_{ℚ} = {_{∑u∈S}λ_{u}u | λ_{u} ≥ 0 in ℚ}.

One new feature is that a strongly convex rational polyedral cone σ has a canonical generating set, constructed as follows. Let ρ be a edge of σ. Since σ is strongly convex, ρ is a ray, i.e., a half-line, and since ρ is rational, the semigroup ρ ∩ N is generated by a unique element u of the intersection. We call u the ray generator of ρ. Figure 6 on the next page shows the ray generator of a rational ray ρ in the plane. The dots are the lattice N = ℤ^{2} and the white ones are ρ ∩ N.

Lemma 2.13. A strongly convex rational polyedral cone is generated by the ray generators of its edges. □

Figure 6. A rational ray ρ ⊆ ℝ^{2} and its unique ray generator

It is customary to call the ray generators of the edges the minimal generators of a strongly convex rational polyhedral cone. Figures 1 and 2 show 3-dimensional strongly convex rational polyhedral cones and their ray generators.

In a similar way, a rational polyhedral cone σ of maximal dimension has unique facet normals, which are the ray generators of the dual σ^{∨}, which is strongly convex by Proposition 2.11.

Here are some especially nice strongly convex cones that play an important role in toric geometry.

Definition 2.14. Let σ ⊆ N_{ℝ} be a strongly convex rational polyhedral cone.

(a) σ is smooth or regular if its minimal generators form part of a ℤ-basis of N,

(b) σ is simplicial if its minimal generators are linearly independent over ℝ.

The cones pictured in Figure 5 are smooth, while those in Figures 1 and 2 are not even simplicial. Note also that the dual of a smooth resp. simplicial cone is again smooth resp. simplicial. Later in the section we will give examples of simplicial cones that are not smooth.

Semigroup Algebras and Affine Toric Varieties. Given a rational polyhedral cone σ ⊆ N_{ℝ}, the lattice points

S_{σ} = σ^{∨} ∩ M ⊆ M

form a semigroup. A key fact is that this semigroup is finitely generated.

Proposition 2.15 (Gordan’s Lemma). S_{σ} = σ^{∨}∩M is finitely generated and hence is an affine semigroup.

Proof. Since σ^{∨} is rational polyhedral, σ^{∨} = Cone(T) for a finite set T ⊆ M. Then K = {∑_{m∈T} δ_{m}m | 0 ≤ δ_{u} < 1} is a bounded region of M_{ℝ} ≃ ℝ^{n}, so that K ∩ M is finite since M ≃ ℤ^{n}. Note that T ∪ (K ∩ M) ⊆ S_{σ}.

We claim T ∪ (K ∩ M) generates S_{σ} as a semigroup. To prove this, take w ∈ S_{σ} and write w = ∑_{m∈T} λ_{m}m where λ_{m} ≥ 0. Then λ_{m} = ⌊λ_{m}⌋ + δ_{m} with ⌊λ_{m}⌋ ∈ ℕ and 0 ≤ δ_{m} < 1, so that

w = ∑_{m∈T}⌊λ_{m}⌋m + ∑_{m∈T} δ_{m}m.

The second sum is in K∩M (remember w ∈ M). It follows that w is a nonnegative integer combination of elements of T ∪ (K ∩ M). □

From §1, we know that affine semigroups give affine toric varieties. Hence we get the following result.

Theorem 2.16. Let σ ⊆ N_{ℝ} ≃ ℝ^{n} be a rational polyhedral cone with semigroup S_{σ} = σ^{∨} ∩ M. Then

U_{σ} = Spec(ℂ[S_{σ}]) = Spec(ℂ[σ^{∨} ∩ M])

is an affine toric variety. Furthermore,

dim U_{σ} = n ⟺ the torus of U_{σ} is T_{N} ⟺ σ is strongly convex.

Proof. By Gordan’s Lemma and Proposition 1.14, U_{σ} is an affine toric variety whose torus has character lattice ℤS_{σ} ⊆ M. To study ℤS_{σ}, note that

ℤS_{σ} = S_{σ} − S_{σ} = {m_{1} − m_{2} | m_{1}, m_{2} ∈ S_{σ}}.

Now suppose that km ∈ ℤS_{σ} for some k > 1 and m ∈ M. Then km = m_{1} − m_{2} for m_{1}, m_{2} ∈ S_{σ} = σ^{∨} ∩ M. Since m_{1} and m_{2} lie in the convex set σ^{∨}, we have

m + m_{2} = \frac{1}{k}m_{1} + \frac{k−1}{k}m_{2} ∈ σ^{∨}.

It follows that m = (m + m_{2}) − m_{2} ∈ ℤS_{σ}, so that M/ℤS_{σ} is torsion-free. Hence

(2.3)  the torus of U_{σ} is T_{N} ⟺ ℤS_{σ} = M ⟺ rank ℤS_{σ} = n.

Since σ is strongly convex if and only if dim σ^{∨} = n (Proposition 2.11), it remains to show that

dim U_{σ} = n ⟺ rank ℤS_{σ} = n ⟺ dim σ^{∨} = n.

The first equivalence follows since the dimension of an affine toric variety is the dimension of its torus, which is the rank of its character lattice. We leave the proof of the second equivalence to the reader (Exercise 2.3). □

Since we want our affine toric varieties to contain the torus T_{N}, we consider only those affine toric varieties U_{σ} for which σ ⊆ N_{ℝ} is strongly convex.

Our first example of Theorem 2.16 is an affine toric variety we know well.

Example 2.17. Let σ = Cone(e_{1}, e_{2}, e_{1} + e_{3}, e_{2} + e_{3}) ⊆ N_{ℝ} = ℝ^{3} with N = ℤ^{3}. This is the cone pictured in Figure 2. By Example 2.8, σ^{∨} is the cone pictured in Figure 1, and by Example 1.17, the lattice points in this cone are generated by columns of matrix (1.6). It follows from Example 1.17 that U_{σ} is the affine toric variety V(xy − zw). ⋄

Here are two further examples of Theorem 2.16.

Example 2.18. Fix 0 ≤ r ≤ n and set σ = Cone(e_{1}, …, e_{r}) ⊆ ℝ^{n}. Then

σ^{∨} = Cone(e_{1}, …, e_{r}, ±e_{r+1}, …, ±e_{n})

and the corresponding affine toric variety is

U_{σ} = Spec(ℂ[x_{1}, …, x_{r}, x_{r+1}^{±1}, …, x_{n}^{±1}]) = ℂ^{r} × (ℂ^{∗})^{n−r}

(Exercise 2.4). This implies the general fact that if σ ⊆ N_{ℝ} ≃ ℝ^{n} is a smooth cone of dimension r, then U_{σ} ≃ ℂ^{r} × (ℂ^{∗})^{n−r}. ⋄

Figure 5 illustrates the cones involved in Example 2.18 when r = 2 and n = 3.

Example 2.19. Fix a positive integer d and let σ = Cone(e_{1}, de_{1} − e_{2}) ⊆ ℝ^{2}. This has dual cone σ^{∨} = Cone(e_{1}, e_{1} + de_{2}). Figure 7 shows σ^{∨} when d = 4. The affine semigroup S_{σ} = σ^{∨} ∩ ℤ^{2} is generated by the lattice points (1, i) for 0 ≤ i ≤ d. When d = 4, these are the white dots in Figure 7. (You will prove these assertions in Exercise 2.5.)

Figure 7. The cone σ^{∨} when d = 4

By §1, the affine toric variety U_{σ} is the Zariski closure of the image of the map Φ : (ℂ^{∗})^{2} → ℂ^{d+1} defined by

Φ(s, t) = (s, st, st^{2}, …, st^{d}).

This map has the same image as the map (s, t) ↦ (s^{d}, s^{d−1}t, …, st^{d−1}, t^{d}) used in Example 1.6. Thus U_{σ} is isomorphic to the rational normal cone Ĉ_{d} ⊆ ℂ^{d+1} whose ideal is generated by the 2 minors of the matrix

(\begin{matrix}x_{0} & x_{1} & ⋯ & x_{d−2} & x_{d−1} \\ x_{1} & x_{2} & ⋯ & x_{d−1} & x_{d} \end{matrix}) .

Note that the cones σ and σ^{∨} are simplicial but not smooth. ⋄

We will return to this example often as we develop the theory of toric varieties. One thing evident in Example 1.6 is the difference between cone generators and semigroup generators: the cone σ^{∨} has two generators but the semigroup S_{σ} = σ^{∨} ∩ ℤ^{2} has d + 1.

When σ ⊆ N_{ℝ} has maximal dimension, the semigroup S_{σ} = σ^{∨} ∩ M has a unique minimal generating set constructed as follows. Define an element m ≠ 0 of S_{σ} to be irreducible if m = m′ + m′′ for m′, m′′ ∈ S_{σ} implies m′ = 0 or m′′ = 0.

Proposition 2.20. Let σ ⊆ N_{ℝ} be strongly convex of maximal dimension and let S_{σ} = σ^{∨} ∩ M. Then

H = {m ∈ S_{σ} | m is irreducible}

has the following properties:

(a) H is finite and generates S_{σ}.

(b) H contains the ray generators of the edges of σ^{∨}.

(c) H is the minimal generating set of S_{σ} with respect to inclusion.

Proof. Proposition 2.11 implies that σ^{∨} is strongly convex, so we can find an element u ∈ σ ∩ N \ {0} such that ⟨m, u⟩ ∈ ℕ for all m ∈ S_{σ} and ⟨m, u⟩ = 0 if and only if m = 0.

Now suppose that m ∈ S is not irreducible. Then m = m′ + m′′ where m′ and m′′ are nonzero elements of S_{σ}. It follows that

⟨m, u⟩ = ⟨m′, u⟩ + ⟨m′′, u⟩

with ⟨m′, u⟩, ⟨m′′, u⟩ ∈ ℕ \ {0}, so that

⟨m′, u⟩ < ⟨m, u⟩ and ⟨m′′, u⟩ < ⟨m, u⟩.

Using induction on ⟨m, u⟩, we conclude that every element of S_{σ} is a sum of irreducible elements, so that H generates S_{σ}. Furthermore, using a finite generating set of S_{σ}, one easily sees that H is finite. This proves part (a). The remaining parts of the proof are covered in Exercise 2.6. □

The set H ⊆ S_{σ} is called the Hilbert basis of S_{σ} and its elements are the minimal generators of S_{σ}. Algorithms for computing Hilbert bases are discussed in [23, Section 7.3] and [33, Chapter 13].

Exercises for § 2.

2.1. Prove Proposition 2.11. 2.2. Let σ ⊆ N_{ℝ} be a polyhedral cone.

(a) Use Proposition 2.9 to prove that σ has a unique minimal face with respect to inclusion. Let W denote this minimal face.

(b) Prove that W = (σ^{∨})^{⊥}.

(c) Prove that W is the largest subspace contained in σ.

2. Cones and Affine Toric Varieties 31

(d) Prove that W = σ ∩ (−σ).

(e) Prove that \overline{σ} = σ/W ⊆ N_{ℝ}/W is a strongly convex polyhedral cone. 2.3. Fix a lattice M ≃ ℝ^{n} and let Span(S) denote the span over ℝ of a subset S ⊆ M_{ℝ}.

(a) Let S ⊆ M be finite. Prove that rank ℤS = dim Span(S).

(b) Let S ⊆ M_{ℝ} be finite. Prove that dim Cone(S) = dim Span(S).

(c) Use parts (a) and (b) to complete the proof of Theorem 2.16. 2.4. Prove the assertions made in Example 2.18. 2.5. Prove the assertions made in Example 2.19. Hint: First show that when a cone is smooth, the ray generators of the cone also generate the corresponding semigroup. Then write the cone σ^{∨} of Example 2.19 as a union of such cones. 2.6. Complete the proof of Proposition 2.20. Hint for part (b): Show that the ray generators of the edges of σ^{∨} are irreducible in S_{σ}. Given an edge ρ of σ^{∨}, it will help to pick u ∈ σ ∩ N \ {0} such that ρ = H_{u} ∩ σ^{∨}. 2.7. Let σ ⊆ N_{ℝ} be a cone generated by a set of linearly independent vectors in N_{ℝ}. Show that σ is strongly convex and simplicial. 2.8. Explain the picture illustrated in Figure 8 in terms of Proposition 2.7.

Figure 8. A cone σ in the plane and its dual

2.9. Let P ⊆ N_{ℝ} be a polytope lying in an affine hyperplane (= translate of a hyperplane) not containing the origin. Generalize Figure 3 by showing that P gives a convex polyhedral cone in N_{ℝ}. Draw a picture. 2.10. Consider the cone σ = Cone(3e_{1} − 2e_{2}, e_{1}) ⊆ ℝ^{2}.

(a) Describe σ^{∨} and find generators of σ^{∨} ∩ ℤ^{2}. Draw a picture similar to Figure 7.

(b) Compute the toric ideal of the affine toric variety U_{σ} and explain how this exercise relates to Exercise 1.6.

2.11. Consider the simplicial cone σ = Cone(e_{1}, e_{2}, e_{1} + e_{2} + 2e_{3}) ⊆ ℝ^{3}.

(a) Describe σ^{∨} and find generators of σ^{∨} ∩ ℤ^{3}.

(b) Compute the toric ideal of the affine toric variety U_{σ}. 2.12. Let σ be a strongly convex polyhedral cone of maximal dimension. Here is an example taken from [13, p. 132] to show that σ and σ^{∨} need not have the same number of edges. Let σ ⊆ ℝ^{4} be the cone generated by 2e_{i} + e_{j} for all 1 ≤ i, j ≤ 4, i ≠ j.

(a) Show that σ has 12 edges.

(b) Show that σ^{∨} is generated by e_{i} and −e_{i} + 2 ∑_{j≠i} e_{j}, 1 ≤ i ≤ 4 and has 8 edges.

3. Properties of Affine Toric Varieties

The final task of this chapter is to explore the properties of affine toric varieties. We will also study maps between affine toric varieties.

Points of Affine Toric Varieties. We first consider various ways to describe the points of an affine toric variety.

Proposition 3.1. Let V = Spec(ℂ[S]) be the affine toric variety of the affine semigroup S. Then there are bijective correspondences between the following objects:

(a) Points p ∈ V.

(b) Maximal ideals 𝔪 ⊆ ℂ[S].

(c) Semigroup homomorphisms S → ℂ, where ℂ is considered as a semigroup under multiplication.

Proof. The correspondence between (a) and (b) is standard (see [7, Theorem 5 of Chapter 5]). The correspondence between (a) and (c) is special to the toric case.

Given a point p ∈ V, define S → ℂ by sending m ∈ S to χ^{m}(p) ∈ ℂ. This makes sense since χ^{m} ∈ ℂ[S] = ℂ[V]. One easily checks that S → ℂ is a semigroup homomorphism.

Going the other way, let γ : S → ℂ be a semigroup homomorphism. Since {χ^{m}}_{m∈S} is a basis of ℂ[S], γ induces a surjective linear map ℂ[S] → ℂ which is a ℂ-algebra homomorphism. The kernel is a maximal ideal and thus gives a point p ∈ V by the correspondence between (a) and (b).

We construct p concretely as follows. Let A = {m_{1}, …, m_{s}} generate S, so that V = Y_{A} ⊆ ℂ^{s}. Let p = (γ(m_{1}), …, γ(m_{s})) ∈ ℂ^{s}. Let us prove that p ∈ V. By Proposition 1.9, it suffices to show that x^{α} − x^{β} vanishes at p for all exponent vectors α = (a_{1}, …, a_{s}) and β = (a_{1}, …, a_{s}) satisfying

∑_{i=1}^{s} a_{i}m_{i} = ∑_{i=1}^{s} b_{i}m_{i}.

This is easy, since γ being a semigroup homomorphism implies that

∏_{i=1}^{s} γ(m_{i})^{a_{i}} = γ( ∑_{i=1}^{s} a_{i}m_{i}) = γ( ∑_{i=1}^{s} b_{i}m_{i}) = ∏_{i=1}^{s} γ(m_{i})^{b_{i}}.

It is straightforward to show that this point of V agrees with the one constructed in the previous paragraph (Exercise 3.1). □

As an application of this result, we describe the torus action on V. In terms of the embedding V = Y_{A} ⊆ ℂ^{s}, the proof of Proposition 1.8 shows that the action of T_{N} on Y_{A} is induced by the usual action of (ℂ^{∗})^{s} on ℂ^{s}. But how do we see the action instrinsically, without embedding into affine space? This is where semigroup homomorphisms prove their value. Fix t ∈ T_{N} and p ∈ V, and let p correspond to the semigroup homorphism m ↦ γ(m). In Exercise 3.1 you will show that t · p is given by the semigroup homomorphism m ↦ χ^{m}(t)γ(m). This abstract description will prove useful in Chapter 3 when we study the orbits of the torus action.

From the point of view of group actions, the action of T_{N} on V is given by a map T_{N} ×V → V. Since both sides are affine varieties, this should be a morphism, meaning that it should come from a ℂ-algebra homomorphism

ℂ[S] = ℂ[V] ⟶ ℂ[T_{N} × V] = ℂ[T_{N}] ⊗_{ℂ} ℂ[V] = ℂ[M] ⊗_{ℂ} ℂ[S].

This homomorphism is given by χ^{m} ↦ χ^{m} ⊗ χ^{m} for m ∈ S (Exercise 3.2).

We can also characterize those affine toric varieties for which the torus action has a fixed point. We say that an affine semigroup S is pointed if S ∩ (−S) = {0}, i.e., if 0 is the only element of S with an inverse. This is the semigroup analog of being strongly convex.

Proposition 3.2. Let V be the affine toric variety. Then:

(a) If we write V = Spec(ℂ[S]), then the torus action has a fixed point if and only if S is pointed, in which case the unique fixed point is given by the the semigroup homomorphism S → ℂ defined by

(3.1)  m ⟼ \begin{cases}1 & m = 0 \\ 0 & m ≠ 0. \end{cases}

(b) If we write V = Y_{A} ⊆ ℂ^{s} for A ⊆ S \ {0}, then the torus action has a fixed point if and only 0 ∈ Y_{A} , in which case the unique fixed point is 0.

Proof. For part (a), let p ∈ V be represented by the semigroup homomorphism γ : S → ℂ. Then p is fixed by the torus action if and only if χ^{m}(t)γ(m) = γ(m) for all m ∈ S and t ∈ T_{N}. This equation is satisfied for m = 0 since γ(0) = 1, and if m ≠ 0, then picking t with χ^{m}(t) ≠ 0 shows that γ(m) = 0. Thus, if a fixed point exists, then it is unique and is given by (3.1). Then we are done since (3.1) is a semigroup homomorphism if and only if S is pointed.

For part (b), first assume that V = Y_{A} ⊆ ℂ^{s} has a fixed point, in which case S = ℕA is pointed and the unique point p is given by (3.1). Then A ⊆ S \ {0} and the proof of Proposition 3.1 imply that p is the origin in ℂ^{s}, so that 0 ∈ Y_{A} . The converse follows since 0 ∈ ℂ^{s} is fixed by (ℂ^{∗})^{s} hence by T_{N} ⊆ (ℂ^{∗})^{s}. □

When V = U_{σ}, we can state Proposition 3.2 as follows (Exercise 3.3).

Corollary 3.3. Let σ ⊆ N_{ℝ} be a strongly convex polyhedral cone. Then the torus action on U_{σ} has a fixed point if and only if dim σ = dim N_{ℝ}, in which case the fixed point is unique and is given by the maximal ideal

⟨χ^{m} | m ∈ S_{σ} \ {0}⟩ ⊆ ℂ[S_{σ}],

where as usual S_{σ} = σ^{∨} ∩ M. □

We will see in Chapter 3 that this corollary is part of the correspondence between torus orbits of U_{σ} and faces of σ.

Normality and Saturation. We next study the question of when an affine toric variety V is normal. We need one definition before stating our normality criterion.

Definition 3.4. An affine semigroup S ⊆ M is saturated if for all k ∈ ℕ \ {0} and m ∈ M, km ∈ S implies m ∈ S.

For example, if σ ⊆ N_{ℝ} is a strongly convex rational polyhedral cone, then S_{σ} = σ^{∨} ∩ M is easily seen to be saturated (Exercise 3.4).

Theorem 3.5. Let V be an affine toric variety with torus T_{N}. Then the following are equivalent:

(a) V is normal.

(b) V = Spec(ℂ[S]), where S ⊆ M is a saturated affine semigroup.

(c) V = Spec(ℂ[S_{σ}]) (= U_{σ}), where S_{σ} = σ^{∨} ∩ M and σ ⊆ N_{ℝ} is a strongly convex rational polyhedral cone.

Proof. By Theorem 1.16, V = Spec(ℂ[S]) for an affine semigroup S contained in a lattice, and by Proposition 1.14, the torus of V has the character lattice M = ℤS. Also let n = dim V, so that M ≃ ℤ^{n}. We will use this to prove (a) ⇒ (b) ⇒ (c).

(a) ⇒ (b): If V is normal, then ℂ[S] = ℂ[V] is integrally closed in its field of fractions ℂ(V). Suppose that km ∈ S for some k ∈ ℕ \ {0} and m ∈ M. Then χ^{m} is a polynomial function on T_{N} and hence a rational function on V since T_{N} ⊆ V is Zariski open. We also have χ^{km} ∈ ℂ[S] since km ∈ S. It follows that χ^{m} is a root of the monic polynomial X^{k} − χ^{km} with coefficients in ℂ[S]. By the definition of normal, we obtain χ^{m} ∈ ℂ[S], i.e., m ∈ S. Thus S is saturated.

(b) ⇒ (c): Let A ⊆ S be a finite generating set of S. Then S lies in the rational polyhedral cone Cone(A) ⊆ M_{ℝ}, and rank ℤA = n implies dim Cone(A) = n by Exercise 2.3. It follows that σ = Cone(A)^{∨} ⊆ N_{ℝ} is a strongly convex rational polyhedral cone such that S ⊆ σ^{∨}∩M. In Exercise 3.4 you will prove that equality holds when S is saturated. Hence S = S_{σ}.

(c) ⇒ (a): We need to show that ℂ[S_{σ}] = ℂ[σ^{∨} ∩ M] is normal when σ ⊆ N_{ℝ} is a strongly convex rational polyhedral cone. Let ρ_{1}, …, ρ_{r} be the rays of σ. Since σ is generated by its rays (Lemma 2.13), we have

σ^{∨} = ⋂_{i=1}^{r} ρ_{i}^{∨},

and intersecting with M gives

S_{σ} = ⋂_{i=1}^{r} S_{ρ_{i}}.

By Exercise 0.5, S_{σ} is normal if each S_{ρ_{i}} is normal, so it suffices to prove that S_{ρ} is normal when ρ is a rational ray in N_{ℝ}. Let u ∈ ρ ∩ N be the ray generator of ρ. Since u is primitive, i.e, \frac{1}{k}u ∉ N for all k > 1, we can find a basis e_{1}, …, e_{n} of N with u = e_{1} (Exercise 3.5). Thus we can assume that ρ = Cone(e_{1}), so that

ℂ[S_{ρ}] = ℂ[x_{1}, x_{2}^{±1}, …, x_{n}^{±1}]

by Example 2.18. But ℂ[x_{1}, …, x_{n}] is normal (it is a UFD), so its localization

ℂ[x_{1}, …, x_{n}]_{x_{2}···x_{n}} = ℂ[x_{1}, x_{2}^{±1}, …, x_{n}^{±1}]

is also normal by Exercise 0.5. This completes the proof. □

Example 3.6. We saw in Example 2.17 that V = V(xy − zw) is the affine toric variety U_{σ} of the cone σ = Cone(e_{1}, e_{2}, e_{1} + e_{3}, e_{2} + e_{3}) pictured in Figure 1. Then Theorem 3.5 implies that V is normal, as claimed in Example 1.5. ⋄

Example 3.7. By Example 2.19, the rational normal cone Ĉ_{d} ⊆ ℂ^{d+1} is the affine toric variety of σ = Cone(de_{1}−e_{2}, e_{1}) ⊆ ℝ^{2} and hence is normal by Theorem 3.5.

It is instructive to view this example using the parametrization

Φ_{A} (s, t) = (s^{d}, s^{d−1}t, …, st^{d−1}, t^{d})

from Example 1.6. Plotting the lattice points in A for d = 2 gives the white squares in Figure 9 (a). These generate the semigroup S = ℕA, and the proof of Theorem 3.5 gives the cone σ^{∨} = Cone(e_{1}, e_{2}), which is the first quadrant in the figure.

Figure 9. Lattice points for the rational normal cone Ĉ_{2}

At first glance, something seems wrong. The affine variety Ĉ_{2} is normal, yet in Figure 9 (a) the semigroup generated by the white squares misses some lattice points in σ^{∨}. This semigroup does not look saturated. How can the affine toric variety be normal?

The problem is that we are using the wrong lattice! Proposition 1.8 tells us to use the lattice ℤA, which gives the white dots and squares in Figure 9 (b). This figure shows that the white squares generate the semigroup of lattice points in σ^{∨}. Hence S is saturated and everything is fine. ⋄

This example points out the importance of working with the correct lattice.

The Normalization of an Affine Toric Variety. The normalization of an affine toric variety is easy to describe. Let V = Spec(ℂ[S]) for an affine semigroup S, so that the torus of the V has character lattice M = ℤS. Let Cone(S) denote the cone of any finite generating set of S and set σ = Cone(S)^{∨} ⊆ N_{ℝ}. In Exercise 3.6 you will prove the following.

Proposition 3.8. The above cone σ is a strongly convex rational polyhedral cone in N_{ℝ} and the inclusion ℂ[S] ⊆ ℂ[σ^{∨} ∩ M] induces a morphism U_{σ} → V that is the normalization map of V. □

The normalization of an affine toric variety of the form Y_{A} is constructed by applying Proposition 3.8 to the affine semigroup ℕA and the lattice ℤA.

Example 3.9. Let A = {(4, 0), (3, 1), (1, 3), (0, 4)} ⊆ ℤ^{2}. Then

Φ_{A} (s, t) = (s^{4}, s^{3}t, st^{3}, t^{4})

parametrizes the surface Y_{A} ⊆ ℂ^{4} considered in Exercise 1.7. This is almost the rational normal cone Ĉ_{4}, except that we have omitted s^{2}t^{2}. Using (2, 2) = \frac{1}{2}((4, 0) + (0, 4)), we see that ℕA is not saturated, so that Y_{A} is not normal.

Applying Proposition 3.8, one sees that the normalization of Y_{A} is Ĉ_{4}. This is an affine variety in ℂ^{5}, and the normalization map is induced by the obvious projection ℂ^{5} → ℂ^{4}. ⋄

In Chapter 3 we will show that the normalization map U_{σ} → V constructed in Proposition 3.8 is one-to-one and onto.

Smooth Affine Toric Varieties. Our next goal is to characterize when an affine toric variety is smooth. Since smooth affine varieties are normal (Proposition 0.9), we need only consider toric varieties U_{σ} coming from strongly convex rational polyhedral cones σ ⊆ N_{ℝ}.

We begin by studying U_{σ} when σ has maximal dimension. Then σ^{∨} is strongly convex, so that S_{σ} = σ^{∨} ∩ M has a Hilbert basis H . Furthermore, Corollary 3.3 tells us that torus action on U_{σ} has a unique fixed point, denoted here by p_{σ} ∈ U_{σ}. The point p_{σ} and the Hilbert basis H are related as follows.

Lemma 3.10. Let σ ⊆ N_{ℝ} be a strongly convex rational polyhedral cone of maximal dimension and let T_{p_{σ}}(U_{σ}) be the Zariski tangent space to the affine toric variety U_{σ} at the above point p_{σ}. Then dim T_{p_{σ}}(U_{σ}) = |H |.

Proof. By Corollary 3.3, the maximal ideal of ℂ[S_{σ}] corresponding to p_{σ} is 𝔪 = ⟨χ^{m} | m ∈ S_{σ} \ {0}⟩. Since {χ^{m}}_{m∈S_{σ}} is basis of ℂ[S_{σ}], we obtain

𝔪 = ⨁_{m≠0} ℂχ^{m} = ⨁_{m irreducible}ℂχ^{m} ⊕ ⨁_{m reducible}ℂχ^{m} = ( ⨁_{m∈H} ℂχ^{m}) ⊕ 𝔪^{2}.

It follows that dim 𝔪/𝔪^{2} = |H |. To relate this to the maximal ideal 𝔪_{U_{σ},p_{σ}} in the local ring O_{U_{σ},p_{σ}}, we use the natural map

𝔪/𝔪^{2} ⟶ 𝔪_{U_{σ},p_{σ}}/𝔪_{U_{σ},p_{σ}}^{2}

which is always an isomorphism (Exercise 3.7). Since T_{p_{σ}}(U_{σ}) is the dual space of 𝔪_{U_{σ},p_{σ}}/𝔪_{U_{σ},p_{σ}}^{2}, we see that dim T_{p_{σ}}(U_{σ}) = |H |. □

The Hilbert basis H of S_{σ} gives U_{σ} = Y_{H} ⊆ ℂ^{s}, where s = |H |. This affine embedding is especially nice. Given any affine embedding U_{σ} ↪ ℂ^{ℓ}, we have dim T_{p_{σ}}(U_{σ}) ≤ ℓ by Lemma 0.6. In other words, dim T_{p_{σ}}(U_{σ}) is a lower bound on the dimension of an affine embedding. Then Lemma 3.10 shows that when σ has maximal dimension, the Hilbert basis of S_{σ} gives the most efficient affine embedding of U_{σ}.

Example 3.11. In Example 2.19, we saw that the rational normal cone Ĉ_{d} ⊆ ℂ^{d+1} is the toric variety coming from σ = Cone(e_{1}, de_{1} − e_{2}) ⊆ ℝ^{2} and that S_{σ} = σ^{∨} ∩ ℤ^{2} is generated by (1, i) for 0 ≤ i ≤ d. These generators form the Hilbert basis of S_{σ}, so that the Zariski tangent space of 0 ∈ Ĉ_{d} has dimension d+1. Hence ℂ^{d+1} in the smallest affine space in which we can embed Ĉ_{d}. ⋄

We now come to our main result about smoothness. Recall from §2 that a rational polyhedral cone is smooth if it can be generated by a subset of a basis of the lattice.

Theorem 3.12. Let σ ⊆ N_{ℝ} be a strongly convex polyhedral cone. Then U_{σ} is smooth if and only if σ is smooth. Furthermore, all smooth affine toric varieties are of this form.

Proof. If an affine toric variety is smooth, then it is normal by Proposition 0.9 and hence of the form U_{σ}. Also, Example 2.18 implies that if σ is smooth as a cone, then U_{σ} is smooth as a variety. It remains to prove the converse. So fix σ ⊆ N_{ℝ} such that U_{σ} is smooth. Let n = dim U_{σ} = dim N_{ℝ}.

First suppose that σ has dimension n and let p_{σ} ∈ U_{σ} be the point studied in Lemma 3.10. Since p_{σ} is smooth in U_{σ}, the Zariski tangent space T_{p_{σ}}(U_{σ}) has dimension n by Definition 0.7. On the other hand, Lemma 3.10 implies that dim T_{p_{σ}}(U_{σ}) is the cardinality of the Hilbert basis H of S_{σ} = σ^{∨} ∩ M. Thus

n = |H | ≥ |{edges ρ ⊆ σ^{∨}}| ≥ n,

where the first inequality holds by Proposition 2.20 (each edge ρ ⊆ σ^{∨} contributes an element of H ) and the second holds since dim σ^{∨} = n. It follows that σ has n edges and H consists of the ray generators of these edges. Since M = ℤS_{σ} by (2.3), the n edge generators of σ^{∨} generate the lattice M ≃ ℤ^{n} and hence form a basis of M. Thus σ^{∨} is smooth, and then σ = (σ^{∨})^{∨} is smooth since duality preserves smoothness.

Next suppose that dim σ = r < n. We reduce to the previous case as follows. Let N_{1} ⊆ N be the smallest saturated sublattice containing the generators of σ. Then N/N_{1} is torsion-free, which by Exercise 3.5 implies the existence of a sublattice N_{2} ⊆ N with N = N_{1} ⊕ N_{2}. Note that rank N_{1} = r and rank N_{2} = n − r.

The cone σ lies in both (N_{1})_{ℝ} and N_{ℝ}. This gives affine toric varieties U_{σ,N_{1}} and U_{σ,N} of dimensions r and n respectively. Furthermore, N = N_{1} ⊕ N_{2} induces M = M_{1} ⊕ M_{2}, so that σ ⊆ (N_{1})_{ℝ} and σ ⊆ N_{ℝ} give the affine semigroups S_{σ,N_{1}} ⊆ M_{1} and S_{σ,N} ⊆ M respectively. It is straighforward to show that

S_{σ,N} = S_{σ,N_{1}} ⊕ M_{2},

which in terms of semigroup algebras can be written

ℂ[S_{σ,N}] ≃ ℂ[S_{σ,N_{1}}] ⊗_{ℂ} ℂ[M_{2}].

The right-hand side is the coordinate ring of U_{σ,N_{1}} × T_{N_{2}}. Thus

(3.2)  U_{σ,N} ≃ U_{σ,N_{1}} × T_{N_{2}},

which in turn implies that

U_{σ,N} ≃ U_{σ,N_{1}} × (ℂ^{∗})^{n−r} ⊆ U_{σ,N_{1}} × ℂ^{n−r}.

Since we are assuming that U_{σ,N} is smooth, it follows that U_{σ,N_{1}}× ℂ^{n−r} is smooth at any point (p, q) in U_{σ,N_{1}} × (ℂ^{∗})^{n−r}. In Exercise 3.8 you will show that

(3.3)  U_{σ,N_{1}} × ℂ^{n−r} is smooth at (p, q) ⟹ U_{σ,N_{1}} is smooth at p.

Letting p = p_{σ} ∈ U_{σ,N_{1}}, the previous case implies that σ is smooth in N_{1} since dim σ = dim(N_{1})_{ℝ}. Hence σ is clearly smooth in N = N_{1} ⊕ N_{2}. □

Equivariant Maps between Affine Toric Varieties. We next study maps V_{1} → V_{2} between affine toric varieties that respect the torus actions on V_{1} and V_{2}.

Definition 3.13. Let V_{i} = Spec(ℂ[S_{i}]) be the affine toric varieties coming from the affine semigroups S_{i}, i = 1, 2. Then a morphism φ : V_{1} → V_{2} is toric if the corresponding map of coordinate rings φ^{∗} : ℂ[S_{2}] → ℂ[S_{1}] is induced by a semigroup homomorphism φ̂: S_{2} → S_{1}.

Here is our first result concerning toric morphisms.

Proposition 3.14. Let T_{N_{i}} be the torus of the affine toric variety V_{i}, i = 1, 2. Then:

(a) A morphism φ : V_{1} → V_{2} is toric if and only if

φ(T_{N_{1}}) ⊆ T_{N_{2}}

and φ|_{T_{N_{1}}} : T_{N_{1}} → T_{N_{2}} is a group homomorphism.

(b) A toric morphism φ : V_{1} → V_{2} is equivariant, meaning that

φ(t · p) = φ(t) · φ(p)

for all t ∈ T_{N_{1}} and p ∈ V_{1}.

Proof. Let V_{i} = Spec(ℂ[S_{i}]), so that the character lattice of T_{N_{i}} is M_{i} = ℤS_{i}. If φ comes from a semigroup homomorphism φ̂ : S_{2} → S_{1}, then φ̂ extends to a group homomorphism φ̂: M_{2} → M_{1} and hence gives a commutative diagram

ℂ[S_{2}] ⟶^{φ^{∗}} ℂ[S_{1}]
↓  ↓
ℂ[M_{2}] ⟶ ℂ[M_{1}].

Applying Spec, we see that φ(T_{N_{1}}) ⊆ T_{N_{2}}, and φ|_{T_{N_{1}}} : T_{N_{1}} → T_{N_{2}} is a group homomorphism since T_{N_{i}} = Hom_{ℤ}(M_{i}, ℂ^{∗}) by Exercise 1.10. Conversely, if φ satisfies these conditions, then φ|_{T_{N_{1}}} : T_{N_{1}} → T_{N_{2}} induces a diagram as above where the bottom map comes from a group homomorphism φ̂ : M_{2} → M_{1}. This, combined with φ^{∗}(ℂ[S_{2}]) ⊆ S_{1}, implies that φ̂ induces a semigroup homomorphism φ̂: S_{2} → S_{1}. This proves part (a) of the proposition.

For part (b), suppose that we have a toric map φ : V_{1} → V_{2}. The action of T_{N_{i}} on V_{i} is given by a morphism Φ_{i} : T_{N_{i}} × V_{i} ⟶ V_{i}, and equivariance means that we have a commutative diagram

\xymatrix{T_{N_{1}} × V_{1} \ar[r]^{Φ_{1}} \ar[d]_{φ|T_{N_{1}}×φ} & V_{1} \ar[d]^{φ} \\ T_{N_{2}} × V_{2} \ar[r]^{Φ_{2}} & V_{2}.}

If we replace V_{i} with T_{N_{i}} in the diagram, then it certainly commutes since φ|_{T_{N_{1}}} is a group homomorphism. Then the whole diagram commutes since T_{N_{1}}× T_{N_{1}} is Zariski dense in T_{N_{1}} × V_{1}. □

We can also characterize toric morphisms between affine toric varieties coming from strongly convex rational polyhedral cones. First note that a homomorphism φ̄ : N_{1} → N_{2} of lattices gives a group homomorphism φ : T_{N_{1}} → T_{N_{2}} of tori. This follows from T_{N_{i}} = N_{i} ⊗_{ℤ} ℂ^{∗}, and one sees that φ is a morphism. Also, tensoring φ̄ with ℝ gives φ̄_{ℝ} : (N_{1})_{ℝ} → (N_{2})_{ℝ}

Here is the result, whose proof we leave to the reader (Exercise 3.9).

Proposition 3.15. Suppose that we have strongly convex rational polyhedral cones σ_{i} ⊆ (N_{i})_{ℝ} and a homomorphism φ̄ : N_{1} → N_{2}. Then φ : T_{N_{1}} → T_{N_{2}} extends to a map of affine toric varieties φ : U_{σ_{1}} → U_{σ_{2}} if and only if φ̄_{ℝ}(σ_{1}) ⊆ σ_{2}. □

In the remainder of the chapter we explore some interesting classes of toric morphisms.

Faces and Affine Open Subsets. Let σ ⊂ N_{ℝ} be a strongly convex rational polyhedral cone and let τ ⊆ σ be a face. Then we can find m ∈ σ^{∨} ∩ M such that τ = H_{m} ∩ σ. This allows us to relate semigroup algebras of σ and τ as follows.

Proposition 3.16. Let τ = H_{m}∩σ, m ∈ σ^{∨}∩M, be a face of σ as above. Then the semigroup algebra ℂ[S_{τ}] = ℂ[τ^{∨} ∩ M] is the localization of ℂ[S_{σ}] = ℂ[σ^{∨} ∩ M] at χ^{m} ∈ ℂ[S_{σ}]. In other words,

ℂ[S_{τ}] = ℂ[S_{σ}]_{χ^{m}}.

Proof. The inclusion τ ⊆ σ implies S_{σ} ⊂ S_{τ}, and since ⟨m, u⟩ = 0 for all u ∈ τ, we have ±m ∈ τ^{∨}. It follows that

S_{σ} + ℤ(−m) ⊆ S_{τ} .

This inclusion is actually an equality, as we now prove. Fix a finite set S ⊂ M with σ = Cone(S) and pick m′ ∈ S_{τ}. Set

C = max_{u∈S}{|⟨m′, u⟩|} ∈ ℕ.

It is straightforward to show that m′ + Cm ∈ S_{σ}. This proves that

S_{σ} + ℤ(−m) = S_{τ} ,

from which ℂ[S_{τ}] = ℂ[S_{σ}]_{χ^{m}} follows immediately. □

We can interpret this in terms of toric morphisms as follows. by Proposition 3.15, the identity map N → N and the inclusion τ ⊆ σ give the toric morphism U_{τ} → U_{σ} that corresponds to the inclusion ℂ[S_{σ}]) ⊆ ℂ[S_{τ}]. By Proposition 3.16,

(3.4) U_{τ} = Spec(ℂ[S_{τ}]) = Spec(ℂ[S_{σ}]_{χ^{m}}) = Spec(ℂ[S_{σ}])_{χ^{m}} = (U_{σ})_{χ^{m}} ⊆ U_{σ}.

In other words, U_{τ} becomes an affine open subset of U_{σ} when τ is a face of σ. This will be useful in Chapters 2 and 3 when we study the local structure of more general toric varieties.

Sublattices of Finite Index and Rings of Invariants. Another interesting class of toric morphisms arises when we keep the same cone but change the lattice. Here is an example we have already seen.

Example 3.17. In Example 3.7 the dual of σ = Cone(e_{1}, e_{2}) ⊆ ℝ^{2} interacts with the lattices shown in Figure 10 on the next page. To make this precise, let us name the lattices involved: the lattices

N′ = ℤ^{2} ⊆ N = {(a/2, b/2) | a, b ∈ ℤ, a + b ≡ 0 mod 2}

have σ ⊆ N′_{ℝ} ⊆ N_{ℝ}, and the dual lattices

M′ = ℤ^{2} ⊇ M = {(a, b) | a, b ∈ ℤ, a + b ≡ 0 mod 2}

Figure 10. Lattice points of σ^{∨} relative to two lattices

have σ^{∨} ⊆ M′_{ℝ} ⊆ M_{ℝ}. Note that duality reverses inclusions and that M and N are indeed dual under dot product. In Figure 10 (a), the black dots in the first quadrant form the semigroup S_{σ,N′} = σ^{∨} ∩ M′, and in Figure 10 (b), the white dots in the first quadrant form S_{σ,N} = σ^{∨} ∩ M.

This gives the affine toric varieties U_{σ,N′} and U_{σ,N}. Clearly U_{σ,N′} = ℂ^{2} since σ is smooth for N′, while Example 3.7 shows that U_{σ,N} is the rational normal cone Ĉ_{2}. The inclusion N′ ⊂ N gives a toric morphism

ℂ^{2} = U_{σ,N′} ⟶ U_{σ,N} = Ĉ_{2}.

Our next task is to find a nice description of this map. ⋄

In general, suppose we have lattices N′ ⊆ N, where N′ has finite index in N, and let σ ⊆ N′_{ℝ} = N_{ℝ} be a strongly convex rational polyhedral cone. Then the inclusion N′ ⊆ N gives the toric morphism

φ : U_{σ,N′} ⟶ U_{σ,N}.

The dual lattices satisfy M′ ⊇ M, so that φ corresponds to the inclusion

ℂ[S_{σ,N′}] = ℂ[σ^{∨} ∩ M′] ⊇ ℂ[σ^{∨} ∩ M] = ℂ[S_{σ,N}]

of semigroup algebras. The idea is to realize ℂ[S_{σ,N}] as a ring of invariants of a group action on ℂ[S_{σ,N′}]. The group in question is the finite abelian group

(3.5)  G = Hom_{ℤ}(M′/M, ℂ^{∗}),

which acts on ℂ[S_{σ,N′}] via

g · χ^{m′} = g([m′]) χ^{m′}.

One can prove that m′ ∈ M′ lies in M if and only if g([m′]) = 1 for all g ∈ G. Hence the ring of invariants

ℂ[S_{σ,N′}]^{G} = {f ∈ ℂ[S_{σ,N′}] | g · f = f for all g ∈ G},

is precisely ℂ[S_{σ,N}], i.e.,

ℂ[S_{σ,N′}]^{G} = ℂ[S_{σ,N}].

In terms of the corresponding affine toric varieties, this has the following geometric consequence.

Proposition 3.18. If N′ has finite index in N and σ ⊆ N′_{ℝ} = N_{ℝ} is a strongly convex rational polyhedral cone, then the finite group G defined in (3.5) acts on U_{σ,N′} and the toric morphism U_{σ,N′} → U_{σ,N} induces a bijection

U_{σ,N′}/G ≃ U_{σ,N}.

Proof. When a finite group G acts on an affine variety V = Spec(R) over ℂ, we get a set V/G of G-orbits as well as an action of G on the coordinate ring R. The key fact is that V/G has a natural structure as an affine variety, namely

V/G ≃ Spec(R^{G}),

where R^{G} is the ring of invariants of the G-action on R. When V = ℂ^{n} and G acts linearly, this is proved in [7, Theorem 10 of Chapter 7]. This proof adapts easily to the case when R = ℂ[S_{σ,N′}] (Exercise 3.10). □

Here is an example of Proposition 3.18.

Example 3.19. In the situation of Example 3.17, one computes that G is the group μ_{2} = {±1} acting on U_{σ,N′} = Spec(ℂ[s, t]) ≃ ℂ^{2} by −1 · (s, t) = (−s, −t). Thus the rational normal cone Ĉ_{2} is the quotient

ℂ^{2}/μ_{2} = U_{σ,N′}/μ_{2} ≃ U_{σ,N} = Ĉ_{2}.

We can see this explicitly as follows. The invariant ring is easily seen to be

ℂ[s, t]^{μ_{2}} = ℂ[s^{2}, st, t^{2}] = ℂ[Ĉ_{2}] ≃ ℂ[x_{0}, x_{1}, x_{2}]/⟨x_{0}x_{2} − x_{1}^{2}⟩,

where the last isomorphism follows from Example 1.6. From the point of view of invariant theory, the generators s^{2}, st, t^{2} of the ring of invariants give a morphism

Φ : ℂ^{2} ⟶ ℂ^{3}, (s, t) → (s^{2}, st, t^{2})

that is constant on μ_{2}-orbits. This map also separates orbits, so it induces

ℂ^{2}/μ_{2} ≃ Φ(ℂ^{2}) = Ĉ_{2},

where the last equality is by Example 1.6. But we can also think about this in terms of semigroups, where the exponent vectors of s^{2}, st, t^{2} give the Hilbert basis of the semigroup S_{σ,N} pictured in Figure 11 (b). Everything fits together very nicely. ⋄

In Exercise 3.11 you will generalize Example 3.19 to the case of the rational normal cone Ĉ_{d} for arbitrary d.

The group G from (3.5) has some interesting properties (Exercise 3.12).

3. Properties of Affine Toric Varieties 43

Proposition 3.20. Let N be a lattice and N′ ⊆ N a sublattice of finite index, with dual lattices M′ ⊇ M. Define G = Hom_{ℤ}(M′/M, ℂ^{∗}) as in (3.5). Then:

(a) There is a perfect bilinear pairing N/N′×M′/M → ℂ^{∗} that gives a canonical isomorphism N/N′ ≃ G.

(b) G is the kernel of the map of tori T_{N′} → T_{N} induced by N′ ⊆ N.

Here is our final example of Proposition 3.18.

Example 3.21. Let σ ⊂ N_{ℝ} ≃ ℝ^{n} be a simplicial cone of dimension n with ray generators u_{1}, …, u_{n}. Then N′ = ∑_{i=1}^{n} ℤu_{i} is a sublattice of finite index. Furthermore, σ is smooth relative to N′, so that U_{σ,N′} = ℂ^{n}. It follows that G = N/N′ acts on ℂ^{n} with quotient

ℂ^{n}/G = U_{σ,N′}/G ≃ U_{σ,N}.

Hence the affine toric variety of a simplicial cone is the quotient of affine space by a finite abelian group. In the literature, varieties like U_{σ,N} are called orbifolds and are said to be ℚ-factorial.

Exercises for § 3.

3.1. Consider the affine toric variety Y_{A} = Spec(ℂ[S]), where A = {m_{1}, …, m_{s}} and S = ℕA. Let γ : S → ℂ be a semigroup homomorphism. In the proof of Proposition 3.1 we showed that p = (γ(m_{1}), …, γ(m_{s})) lies in Y_{A}.

(a) Prove that the maximal ideal {f ∈ ℂ[S] | f(p) = 0} is the kernel of the ℂ-algebra homomophism ℂ[S] → ℂ induced by γ.

(b) The torus T_{N} of Y_{A} has character lattice M = ℤA and fix t ∈ T_{N}. As in the discussion following Proposition 3.1, this gives the semigroup homomorphism m ↦ χ^{m}(t)γ(m). Prove that this corresponds to the point

(χ^{m_{1}}, …, χ^{m_{s}}) · (γ(m_{1}), …, γ(m_{s})) = (χ^{m_{1}}γ(m_{1}), …, χ^{m_{s}}γ(m_{s}))

coming from the action of t ∈ T_{N} ⊆ (ℂ^{∗})^{s} on p ∈ Y_{A} ⊆ ℂ^{s}.

3.2. Let V = Spec(ℂ[S]) with for torus T_{N} = Spec(ℂ[M]), M = ℤS. The action T_{N} × V → V comes from a ℂ-algebra homomorphism ℂ[S] → ℂ[M] ⊗_{ℂ} ℂ[S]. Prove that this homomorphism is given by χ^{m} ↦ χ^{m} ⊗ χ^{m}. Hint: First show that this formula determines the ℂ-algebra homomorphism ℂ[M] → ℂ[M] ⊗_{ℂ} ℂ[M] that gives the group operation T_{N} × T_{N} → T_{N}.

3.3. Prove Corollary 3.3.

3.4. Let A ⊆ M be a finite set.

(a) Prove that the semigroup ℕA is saturated in M if and only if ℕA = Cone(A) ∩ M. Hint: Apply (2.2) to Cone(A) ⊆ M_{ℝ}.

(b) Complete the proof of (b) ⇒ (c) from Theorem 3.5.

3.5. Let N be a lattice.

(a) Let N_{1} ⊆ N be a sublattice such that N/N_{1} is torsion-free. Prove that there is a sublattice N_{2} ⊆ N such that N = N_{1} ⊕ N_{2}.

44 1. Affine Toric Varieties

(b) Let u ∈ N be primitive as defined in the proof of Theorem 3.5. Prove that N has a basis e_{1}, …, e_{n} such that e_{1} = u.

3.6. Prove Proposition 3.8.

3.7. Let p be a point of an irreducible affine variety V. Then p gives the maximal ideal 𝔪 = {f ∈ ℂ[V] | f(p) = 0} as well as the maximal ideal 𝔪_{V,p} ⊆ O_{V,p} defined in §0. Prove that the natural map 𝔪/𝔪^{2} ⟶ 𝔪_{V,p}/𝔪_{V,p}^{2} is an isomorphism of ℂ-vector spaces.

3.8. Prove (3.3). Hint: Use Lemma 0.6 and Example 0.10.

3.9. Prove Proposition 3.15.

3.10. Prove Proposition 3.18.

3.11. The group of dth roots of unity μ_{d} = {ζ ∈ ℂ^{∗} | ζ^{d} = 1} be acts on ℂ^{2} by ζ · (x, y) = (ζx, ζy). Adapt Example 3.19 to show that ℂ^{2}/μ_{d} ≃ Ĉ_{d}. Hint: Use lattices N′ = ℤ^{2} ⊆ N = {(a/d, b/d) | a, b ∈ ℤ, a + b ≡ 0 mod d}.

3.12. Prove Proposition 3.20. Hint: Use the pairing ⟨ , ⟩ : M_{ℚ} × N_{ℚ} → ℚ and note that M ⊆ M′ ⊆ M_{ℚ} and N′ ⊆ N ⊆ N_{ℚ}. The map ℚ/ℤ → ℂ^{∗} given by [q] ↦ e^{2πiq} will also be helpful.

3.13. Let σ_{1} ⊂ (N_{1})_{ℝ} and σ_{2} ⊂ (N_{2})_{ℝ} be strongly convex rational polyhedral cones. This gives the cone σ_{1} × σ_{2} ⊂ (N_{1} ⊕ N_{2})_{ℝ}. Prove that U_{σ_{1}×σ_{2}} ≃ U_{σ_{1}} × U_{σ_{2}}. Also explain how this result applies to (3.2).

3.14. By Proposition 3.1, a point p of an affine toric variety V = Spec(ℂ[S]) is represented by a semigroup homomorphism γ : S → ℂ. Prove that p lies in the torus of V if and only if γ never vanishes, i.e., γ(m) ≠ 0 for all m ∈ S.

Projective Toric Varieties

0. Background: Projective Varieties

Our discussion assumes that the reader is familiar with the elementary theory of projective varieties, at the level of [7, Chapter 8].

In Chapter 1, we introduced affine toric varieties. In general, a toric variety is an irreducible variety X over ℂ containing a torus T_{N} ≃ (ℂ^{∗})^{n} as a Zariski open set such that the action of (ℂ^{∗})^{n} on itself extends to an action on X. We will learn in Chapter 3 that the concept of “variety” is somewhat subtle. Hence we will defer the formal definition of toric variety until then and instead concentrate on toric varieties that live in projective space ℙ^{n}, defined by

(0.1)  ℙ^{n} = (ℂ^{n+1} \ {0})/ℂ^{∗},

where ℂ^{∗} acts via homotheties, i.e., λ · (a_{0}, …, a_{n}) = (λa_{0}, …, λa_{n}) for λ ∈ ℂ^{∗} and (a_{0}, …, a_{n}) ∈ ℂ^{n+1}. Thus (a_{0}, …, a_{n}) are homogeneous coordinates of a point in ℙ^{n} and are well-defined up to homothety.

The goal of this chapter is to use from lattice points, polytopes, and polyhedra to create toric varieties that lie in ℙ^{n}.

Homogeneous Coordinate Rings. A projective variety V ⊆ ℙ^{n} is defined by the vanishing of finitely many homogeneous polynomials in the polynomial ring S = ℂ[x_{0}, …, x_{n}]. The homogeneous coordinate ring of V is the quotient ring

ℂ[V] = S/I(V),

where I(V) is generated by all homogeneous polynomials that vanish on V.

The polynomial ring S is graded by setting deg(x_{i}) = 1. This gives the decomposition S = ⨁_{d=0}^{∞} S_{d}, where S_{d} is the vector space of homogeneous polynomials of degree d. Homogeneous ideals decompose similarly, and the above coordinate ring ℂ[V] inherits a grading where

ℂ[V]_{d} = S_{d}/I(V)_{d}.

The ideal V(I) ⊆ S = ℂ[x_{0}, …, x_{n}] also defines an affine variety V̂ ⊆ ℂ^{n+1}, called the affine cone of V. The variety V̂ satisfies

(0.2)  V = (V̂ \ {0})/ℂ^{∗},

and its coordinate ring is the homogeneous coordinate ring of V, i.e.,

ℂ[V̂] = ℂ[V].

Example 0.1. In Example 1.6 of Chapter 1 we encountered ideal ideal

I = ⟨x_{i}x_{j+1} − x_{i+1}x_{j} | 0 ≤ i < j ≤ d − 1⟩ ⊆ ℂ[x_{0}, …, x_{d}]

generated by the 2 × 2 minors of the matrix

(\begin{matrix}x_{0} & x_{1} & ⋯ & x_{d−2} & x_{d−1} \\ x_{1} & x_{2} & ⋯ & x_{d−1} & x_{d} \end{matrix}) .

Since I is homogeneous, it defines a projective variety C_{d} ⊆ ℙ^{d} that is the image of the map

Φ : ℙ^{1} ⟶ ℙ^{d}

defined in homogeneous coordinates by (s, t) ↦ (s^{d}, s^{d−1}t, …, st^{d−1}, t^{d}) (see Exercise 1.1 of Chapter 1). This shows that C_{d} is a curve, called the rational normal curve of degree d. Furthermore, the affine cone of C_{d} is the rational normal cone Ĉ_{d} ⊆ ℂ^{d+1} discussed in Example 1.6 of Chapter 1.

We know from Chapter 1 that Ĉ_{d} is an affine toric surface; we will soon see that C_{d} is a projective toric curve. ⋄

Example 0.2. The affine toric variety V(xy − zw) ⊆ ℂ^{4} studied in Chapter 1 is the affine cone of the rojective surface V = V(xy − zw) ⊆ ℙ^{3}. Recall that this surface is isomorphic to ℙ^{1} × ℙ^{1} via the Segre embedding

ℙ^{1} × ℙ^{1} ⟶ ℙ^{3}

given by (s, t; u, v) ↦ (su, tv, sv, tu). We will see below that V ≃ ℙ^{1} × ℙ^{1} is the projective toric variety coming from the unit square in the plane. ⋄

As in the affine case, a projective variety V ⊆ ℙ^{n} has the classical topology, induced from the usual topology on ℙ^{n}, and the Zariski topology, where the Zariski closed sets are subvarieties of V (meaning projective varieties of ℙ^{n} contained in V) and the Zariski open sets are their complements.

Rational Functions on Irreducible Projective Varieties. A homogeneous polynomial f ∈ S of degree d does not give a function on ℙ^{n} since

f(λx_{0}, …, λx_{n}) = λ^{d}f(x_{0}, …, x_{n}).

However, the quotient of two such polynomials f, g ∈ S_{d} gives the well-defined function

\frac{f}{g} : ℙ^{n} \ V(g) → ℂ.

provided g ≠ 0. We write this as f/g : ℙ^{n} −−→ ℂ and say that f/g is a rational function on ℙ^{n}.

More generally, suppose that V ⊆ ℙ^{n} is irreducible, and let f, g ∈ ℂ[V] = ℂ[V̂] be homogeneous of the same degree with g ≠ 0. Then f and g give functions on the affine cone V̂ and hence an element f/g ∈ ℂ(V̂). By (0.2), this induces a rational function f/g : V −−→ ℂ. Thus

ℂ(V) = {f/g ∈ ℂ(V̂) | f, g ∈ ℂ[V] homogeneous of the same degree, g ≠ 0}

is the field of rational functions on V. It is customary to write the set on the left as ℂ(V̂)_{0} since it consists of the degree 0 elements of ℂ(V̂).

Affine Pieces of Projective Varieties. A projective variety V ⊆ ℙ^{n} is a union of Zariski open sets that are affine. To see why, let U_{i} = ℙ^{n} \ V(x_{i}). Then U_{i} ≃ ℂ^{n} via the map

(0.3)  (a_{0}, …, a_{n}) ⟼ (a_{0}/a_{i}, …, a_{i−1}/a_{i}, a_{i+1}/a_{i}, …, a_{n}/a_{i})

so that in the notation of Chapter 1, we have

U_{i} = Spec(ℂ[x_{0}/x_{i}, …, x_{i−1}/x_{i}, x_{i+1}/x_{i}, …, x_{n}/x_{i}]).

Then V ∩ U_{i} is a Zariski open subset of V that maps via (0.3) to the affine variety in ℂ^{n} defined by the equations

(0.4)  f(x_{0}/x_{i}, …, x_{i−1}/x_{i}, 1, x_{i+1}/x_{i}, …, x_{n}/x_{i}) = 0

as f varies over all homogeneous polynomials in I(V).

We call V ∩ U_{i} an affine piece of V. These affine pieces cover V since the U_{i} cover ℙ^{n}. When V is irreducible, we can describe the coordinate rings of the affine pieces as follows. The variable x_{i} induces an element x̄_{i} ∈ ℂ[V] = ℂ[V̂] ⊆ ℂ(V̂), so that we get the localization

(0.5)  ℂ[V]x̄_{i} = {f/x̄_{i}^{k} | f ∈ ℂ[V], k ≥ 0} ⊆ ℂ(V̂)

as in equation (0.2) of Chapter 1. Then

(0.6)  (ℂ[V]x̄_{i})_{0} = {f/x̄_{i}^{k} ∈ ℂ[V]x̄_{i} | f is homogeneous of degree k}

is the subring of ℂ[V]x̄_{i} consisting of all element of degree 0. Note that (0.6) uses only the homogeneous coordinate ring ℂ[V] and the element x̄_{i}. Yet it gives an affine piece of V as follows.

Lemma 0.3. Assume that V is irreducible and define U_{i} and (ℂ[V]x̄_{i})_{0} as above. Then the affine piece V ∩ U_{i} of V has coordinate ring

ℂ[V ∩ U_{i}] ≃ (ℂ[V]x̄_{i})_{0}.

Proof. We have an exact sequence

0 ⟶ I(V) ⟶ ℂ[x_{0}, …, x_{n}] ⟶ ℂ[V] ⟶ 0.

If we localize at x_{i} as in (0.5), we get

0 ⟶ I(V)_{x_{i}} ⟶ ℂ[x_{0}, …, x_{n}]_{x_{i}} ⟶ ℂ[V]x̄_{i} ⟶ 0

which is exact since localization preserves exactness (see Exercise 0.1). Taking elements of degree 0 gives the exact sequence

0 ⟶ (I(V)_{x_{i}})_{0} ⟶ (ℂ[x_{0}, …, x_{n}]_{x_{i}})_{0} ⟶ (ℂ[V]x̄_{i})_{0} ⟶ 0.

Note that (ℂ[x_{0}, …, x_{n}]_{x_{i}})_{0} = ℂ[x_{0}/x_{i}, …, x_{i−1}/x_{i}, x_{i+1}/x_{i}, …, x_{n}/x_{i}]). If f ∈ I(V) is homogeneous of degree k, then

f/x_{i}^{k} = f(x_{0}/x_{i}, …, x_{i−1}/x_{i}, 1, x_{i+1}/x_{i}, …, x_{n}/x_{i}) ∈ (I(V)_{x_{i}})_{0}.

Thus (I(V)_{x_{i}})_{0} comes from the defining equations (0.4) of V ∩ U_{i}. Since ℂ[V] is an integral domain, the same is true for (ℂ[V]x̄_{i})_{0}. It follows that (I(V)_{x_{i}})_{0} is a prime ideal, which by the Nullstellensatz implies that (I(V)_{x_{i}})_{0} = I(V ∩ U_{i}). The lemma follows immediately. □

One can also explore what happens when we intersect affine pieces V ∩ U_{i} and V ∩ U_{i} for i ≠ j. By Exercise 0.2, V ∩ U_{i} ∩ U_{j} is affine with coordinate ring

(0.7)  ℂ[V ∩ U_{i} ∩ U_{j}] ≃ (ℂ[V]_{x_{i}x_{j}})_{0}

when V is irreducible. Actually, irreducibility is not essential here—(0.7) and Lemma 0.3 hold for arbitrary varieties, provided one uses the more general notion of localization mentioned in Exercise 0.2 of Chapter 1.

We will use these results in §2 when we explore the structure of projective toric varieties. We will also see later in the book that Lemma 0.3 is related to the “Proj” construction, where Proj of a graded ring gives a projective variety, just as Spec of an ordinary ring gives an affine variety.

Products of Projective Spaces. One can study the product ℙ^{n} × ℙ^{m} of projective spaces using the bigraded ring ℂ[x_{0}, …, x_{m}, y_{0}, …, y_{m}], where x_{i} has bidegree (1, 0) and y_{i} has bidegree (0, 1). Then a bihomogeneous polynomial f of bidegree (a, b) gives a well-defined equation f = 0 in ℙ^{n} × ℙ^{m}. This allows us to define varieties in ℙ^{n} × ℙ^{m} using bihomogeneous ideals. In particular, the ideal I(V) of a variety V ⊆ ℙ^{n} × ℙ^{m} is a bihomogeneous ideal.

Another way to study ℙ^{n} × ℙ^{m} is via the Segre embedding

ℙ^{n} × ℙ^{m} ⟶ ℙ^{nm+n+m}

defined by mapping (a_{0}, …, a_{n}, b_{0}, …, b_{m}) to the point

(a_{0}b_{0}, a_{0}b_{1}, …, a_{0}b_{m}, a_{1}b_{0}, …, a_{1}b_{m}, a_{n}b_{m}, a_{1}b_{0}, …, a_{n}b_{m}).

This map is studied in [7, Exercise 14 to Chapter 8, §4]. If ℙ^{nm+n+m} has homogeneous coordinates x_{ij} for 0 ≤ i ≤ n, 0 ≤ j ≤ m, then ℙ^{n} × ℙ^{m} ⊆ ℙ^{nm+n+m} is define by vanishing of the 2×2 minors of the (n+1)×(m+1) matrix of variables

(\begin{matrix}x_{00} & ⋯ & x_{0m} \\ ⋮ & & ⋮ \\ x_{n0} & ⋯ & x_{nm} \end{matrix}) .

This follows since an (n + 1) × (m + 1) matrix has rank 1 if and only if it is a product A^{t}B, where A and B are nonzero row matrices of lengths n+1 and m+1.

These approaches give the same notion of what it means to be a subvariety of ℙ^{n}×ℙ^{m}. A homogeneous polynomial F (x_{ij}) of degree d gives the bihomogeneous polynomial F (x_{i}y_{j}) of bidegree (d, d). Hence any subvariety of ℙ^{nm+n+m} lying in ℙ^{n}×ℙ^{m} can be defined by a bihomogeneous ideal in ℂ[x_{0}, …, x_{m}, y_{0}, …, y_{m}]. Going the other way takes more thought and is discussed in Exercise 0.4.

We also have the following useful result proved in Exercise 0.5.

Proposition 0.4. Given subvarieties V ⊆ ℙ^{n} and W ⊆ ℙ^{m}, the product V × W is a subvariety of ℙ^{n} × ℙ^{m}. □

Weighted Projective Space. The graded ring associated to ℙ^{n} is ℂ[x_{0}, …, x_{n}], where each variable x_{1} has degree 1. More generally, let q_{0}, …, q_{n} be positive integers with gcd(q_{0}, …, q_{n}) = 1 and define

ℙ(q_{0}, …, q_{n}) = (ℂ^{n+1} \ {0})/ ∼

where ∼ is the equivalence relation

(a_{0}, …, a_{n}) ∼ (b_{0}, …, b_{n}) ⟺ a_{i} = λ^{q_{i}}b_{i}, i = 0, …, n for some λ ∈ ℂ^{∗}.

We call ℙ(q_{0}, …, q_{n}) a weighted projective space. Note that ℙ^{n} = ℙ(1, …, 1).

The ring corresponding to ℙ(q_{0}, …, q_{n}) is the graded ring ℂ[x_{0}, …, x_{n}] where x_{i} now has degree q_{i}. A polynomial f is weighted homogeneous of degree d if every monomial x^{α} appearing in f satisfies α · (q_{0}, …, q_{n}) = d. The f = 0 is welldefined on ℙ(q_{0}, …, q_{n}) when f is weighted homogeneous, so that one can define varieties in ℙ(q_{0}, …, q_{n}) using weighted homogeneous ideals of ℂ[x_{0}, …, x_{n}].

Example 0.5. We can embed the weighted projective plane ℙ(1, 1, 2) in ℙ^{3} using the monomials x_{0}^{2}, x_{0}x_{1}, x_{1}^{2}, x_{2} of weighted degree 2. In other words, the map

ℙ(1, 1, 2) ⟶ ℙ^{3}

given by

(a_{0}, a_{1}, a_{2}) ⟼ (a_{0}^{2}, a_{0}a_{1}, a_{1}^{2}, a_{2})

50 2. Projective Toric Varieties

is well-defined and injective. One can check that this map induces

ℙ(1, 1, 2) ≃ V(y_{0}y_{2} − y_{1}^{2}) ⊆ ℙ^{3},

where y_{0}, y_{1}, y_{2}, y_{3} are homogeneous coordinates on ℙ^{3}. ⋄

Later in the book we will use toric methods to construct projective embeddings of arbitrary weighted projective spaces.

Exercises for § 0.

0.1. This exercise is concerned with localization over an integral domain R.

(a) Let f ∈ R \ {0} and let 0 → M_{1} → M_{2} → M_{3} be an exact sequence of R-modules. Prove that

0 ⟶ M_{1} ⊗_{R} R_{f} ⟶ M_{2} ⊗_{R} R_{f} ⟶ M_{3} ⊗_{R} R_{f} ⟶ 0

is exact, where R_{f} is the localization of R at f defined in (0.2) of Chapter 1.

(b) Now let R = ℂ[x_{0}, …, x_{n}] and let ℂ[V] = R/I(V) for an irreducible projective variety V ⊆ ℙ^{n}. Also let x̄_{i} be the image of x_{i} in ℂ[V]. Note the ℂ[V] is an Rmodule. Prove that ℂ[V]x̄_{i} ≃ ℂ[V] ⊗_{R} R_{x_{i}}.

0.2. Prove the claim made in (0.7).

0.3. Let V ⊆ ℙ^{n} be a projective variety. Take f_{0}, …, f_{m} ∈ S_{d} such that the intersection V ∩ V(f_{0}, …, f_{m}) is empty. Prove that the map

(a_{0}, …, a_{n}) ⟼ (f_{0}(a_{0}, …, a_{n}), …, f_{m}(a_{0}, …, a_{n}))

induces a well-defined map function Φ : V → ℙ^{m}.

0.4. Let V ⊆ ℙ^{n} × ℙ^{m} be defined by f_{ℓ}(x_{i}, y_{j}) = 0, where f_{ℓ}(x_{i}, y_{j}) is bihomogenous of bidegree (a_{ℓ}, b_{ℓ}), ℓ = 1, …, s. The goal of this exercise is to show that when we embed ℙ^{n} × ℙ^{m} in ℙ^{nm+n+m} via the Segre embedding described in the text, V becomes a subvariety of ℙ^{nm+n+m}.

(a) For each ℓ, pick an integer d_{ℓ} ≥ max{a_{ℓ}, b_{ℓ}} and consider the polynomials f_{ℓ,α,β} = x^{α}y^{β}f_{i}(x_{i}, y_{j}) where ℓ = 0, …, s and |α| = d_{ℓ} − a_{ℓ}, |β| = d_{ℓ} − b_{ℓ}. Note that f_{ℓ,α,β} is bihomogenous of bidegree (d_{ℓ}, d_{ℓ}). Prove that V ⊆ ℙ^{n} × ℙ^{m} is defined by the vanishing of the f_{ℓ,α,β}.

(b) Use part (a) to show that V is a subvariety of ℙ^{nm+n+m} under the Segre embedding.

0.5. Prove Proposition 0.4

0.6. Consider the Segre embedding ℙ^{1} ×ℙ^{1} → ℙ^{3}. Show that after relabeling coordinates, the affine cone of ℙ^{1}×ℙ^{1} in ℙ^{3} is the variety V(xy−zw) ⊆ ℂ^{4} featured in many examples in Chapter 1.

1. Lattice Points and Projective Toric Varieties

We first observe that ℙ^{n} is a toric variety with torus

T_{ℙ^{n}} = ℙ^{n} \ V(x_{0} … x_{n}) = {(a_{0}, …, a_{n}) ∈ ℙ^{n} | a_{0} … a_{n} ≠ 0}
= {(1, t_{1}, …, t_{n}) ∈ ℙ^{n} | t_{1}, …, t_{n} ∈ ℂ^{∗}} ≃ (ℂ^{∗})^{n}.

The action of T_{ℙ^{n}} on itself clearly extends to an action on ℙ^{n}, making ℙ^{n} a toric variety. To describe the lattices associated to T_{ℙ^{n}}, we use the exact sequence of tori

1 ⟶ ℂ^{∗} ⟶ (ℂ^{∗})^{n+1} ⟶^{π} T_{ℙ^{n}} ⟶ 1

coming from the definition (0.1) of ℙ^{n}. Hence the character lattice of T_{ℙ^{n}} is

(1.1)  M_{n} = {(a_{0}, …, a_{n}) ∈ ℤ^{n+1} | ∑_{i=0}^{n}a_{i} = 0},

and the lattice of one-parameter subgroups N_{n} is the quotient

N_{n} = ℤ^{n+1}/ℤ(1, …, 1).

Lattice Points and Projective Toric Varieties. Let T_{N} be a torus with lattices M and N as usual. In Chapter 1, we used a finite set of lattice points of A = {m_{1}, …, m_{s}} ⊆ M to create the affine toric variety Y_{A} as the Zariski closure of the image of the map

Φ_{A} : T_{N} ⟶ ℂ^{s}, t ⟼ (χ^{m_{1}}(t), …, χ^{m_{s}}(t)).

To get a projective toric variety, we regard Φ_{A} as a map to (ℂ^{∗})^{s} and compose with the homomorphism π : (ℂ^{∗})^{s} → T_{ℙ^{s−1}} to obtain

(1.2)  T_{N} ⟶^{Φ_{A}} ℂ^{s} ⟶^{π} T_{ℙ^{s−1}} ⊆ ℙ^{s−1}.

Definition 1.1. Given a finite set A ⊆ M, the projective toric variety X_{A} is defined to be the Zariski closure in ℙ^{s−1} of the image of the map Φ_{A} from (1.2).

Proposition 1.2. X_{A} is a toric variety of dimension equal to the dimension of the smallest affine subspace of M_{ℝ} containing A.

Proof. The proof that X_{A} ⊆ ℙ^{s−1} is a toric variety is similar to the proof given in Propostion 1.8 of Chapter 1 that Y_{A} ⊆ ℙ^{s} is a toric variety. The assertion concerning the dimension of X_{A} will follow from Proposition 1.6 below. □

More concretely, X_{A} is the Zariski closure of the image of the map

T_{N} ⟶ ℙ^{s−1}, t ⟼ (χ^{m_{1}}(t), …, χ^{m_{s}}(t))

given by the characters coming from A = {m_{1}, …, m_{s}} ⊆ M. In particular, if M = ℤ^{n}, the χ^{m_{i}} is the Laurent monomial t^{m_{i}} and X_{A} as the Zariski closure of the image of

T_{N} ⟶ ℙ^{s−1}, t ⟼ (t^{m_{1}}, …, t^{m_{s}}).

In the literature, A ⊆ ℤ^{n} is often given as an n × s integer matrix, so that the elements of A are the columns of the matrix.

Here is a example where the lattice points themselves are matrices.

Example 1.3. Let M = ℤ^{3×3} be the lattice of 3 × 3 integer matrices and let

P_{3} = {3 × 3 permutation matrices} ⊆ ℤ^{3×3}.

Write ℂ[M]) = ℂ[t_{1}^{±1}, …, t_{9}^{±1}], where the variables give the generic 3 × 3 matrix

(\begin{matrix}t_{1} & t_{2} & t_{3} \\ t_{4} & t_{5} & t_{6} \\ t_{7} & t_{8} & t_{9} \end{matrix})

with nonzero entries. Also let ℙ^{5} have homogeneous coordinates x_{ijk} indexed by triples such that ( \begin{matrix}1 & 2 & 3 \\ i & j & k \end{matrix} ) is a permutation in S_{3}. Then X_{P_{3}} ⊆ ℙ^{5} is the Zariski closure of the image of the map T_{N} → ℙ^{5} given by the Laurent monomials t_{i}t_{j}t_{k} for ( \begin{matrix}1 & 2 & 3 \\ i & j & k \end{matrix} ) ∈ S_{3}. The ideal of X_{P_{3}} is

I(X_{P_{3}} ) = ⟨x_{123}x_{231}x_{312} − x_{132}x_{321}x_{213}⟩ ⊆ ℂ[x_{ijk}],

where the relation comes from the fact that three of the permutation matrices sum to the other three (Exercise 1.1). Ideals of the toric varieties arising from permutation matrices have applications to sampling problems in statistics [33, p. 148]. ⋄

The Affine Cone of a Projective Toric Variety. The projective variety X_{A} ⊆ ℙ^{s−1} has an affine cone X̂_{A} ⊆ ℂ^{s}. How does X̂_{A} relate to the affine toric variety Y_{A} ⊆ ℂ^{s} constructed in Chapter 1?

Recall from Chapter 1 that when A = {m_{1}, …, m_{s}} ⊆ M, the map e_{i} ↦ m_{i} induces an exact sequence

(1.3)  0 ⟶ L ⟶ ℤ^{s} ⟶ M

and that the ideal of Y_{A} is the toric ideal

I_{L} = ⟨x^{α} − x^{β} | α, β ∈ ℕ^{s} and α − β ∈ L⟩

(Proposition 1.9 of Chapter 1). Then we have the following result.

Proposition 1.4. Given Y_{A} , X_{A} and I_{L} as above, the following are equivalent:

(a) Y_{A} ⊆ ℂ^{s} is the affine cone X̂_{A} of X_{A} ⊆ ℙ^{s−1}.

(b) I_{L} = I(X_{A} ).

(c) I_{L} is homogeneous.

(d) There is u ∈ N and k > 0 in ℕ such that ⟨m_{i}, u⟩ = k for i = 1, …, s.

Proof. The equivalence (a) ⇔ (b) follows from the equalities I(X_{A} ) = I(X̂_{A} ) and I_{L} = I(Y_{A} ), and the implication (b) ⇒ (c) is obvious.

For (c) ⇒ (d), assume that I_{L} is homogeneous and take x^{α} − x^{β} ∈ L for α − β ∈ L. If x^{α} and x^{β} had different degrees, then x^{α}, x^{β} ∈ I_{L} = I(Y_{A} ) would vanish on Y_{A} . This is impossible since (1, …, 1) ∈ Y_{A} by (1.2). Hence x^{α} and x^{β} have the same degree, which implies that ℓ · (1, …, 1) = 0 for all ℓ ∈ L. Now tensor (1.3) with ℚ and take duals to obtain an exact sequence

N_{ℚ} ⟶ ℚ^{s} ⟶ Hom_{ℚ}(L_{ℚ}, ℚ) ⟶ 0.

The above argument shows that (1, …, 1) ∈ ℚ^{s} maps to zero in Hom_{ℚ}(L_{ℚ}, ℚ) and hence comes from an element ũ ∈ N_{ℚ}. In other words, ⟨m_{i}, ũ⟩ = 1 for all i. Clearing denominators gives the desired u ∈ N and k > 0 in ℕ.

Finally, we prove (d) ⇒ (b). Since I_{L} = I(Y_{A} ), it suffices to show that

X̂_{A} ∩ (ℂ^{∗})^{s} ⊆ Y_{A} .

Let p ∈ X̂_{A} ∩ (ℂ^{∗})^{s}. Since X_{A} ∩ T_{ℙ^{s−1}} is the torus of X_{A} , it follows that

p = μ · (χ^{m_{1}}(t), …, χ^{m_{s}}(t))

for some μ ∈ ℂ^{∗} and t ∈ T_{N}. Let u ∈ N be as in part (d). This gives a oneparameter subgroup of T_{N}, which we write as τ ↦ λ^{u}(τ) for τ ∈ ℂ^{∗}. Then λ^{u}(τ)t ∈ T_{N} maps to the point q ∈ Y_{A} given by

q = (χ^{m_{1}}(λ^{u}(τ)t), …, χ^{m_{s}}(λ^{u}(τ)t)) = (τ^{⟨m_{1},u⟩}χ^{m_{1}}(t), …, τ^{⟨m_{s},u⟩}χ^{m_{s}}(t)),

since χ^{m}(λ^{u}(τ)) = τ^{⟨m,u⟩} by the description of ⟨ , ⟩ given in §1 of Chapter 1. The hypothesis of part (d) allows us to rewrite q as

q = τ^{k} · (χ^{m_{1}}(t), …, χ^{m_{s}}(t)).

Using k > 0, we can choose τ so that p = q ∈ Y_{A} . This completes the proof. □

The condition ⟨m_{i}, u⟩ = k, i = 1, …, s for some u ∈ N and k > 0 in ℕ means that A lies in an affine hyperplane of M_{ℚ} not containing the origin. When M = ℤ^{n} and A consists of the columns of an n × s integer matrix, this condition is equivalent to (1, …, 1) lying in the row space of the matrix (Exercise 1.2).

Example 1.5. We will examine the rational normal curve C_{d} ⊆ ℙ^{d} using two different sets of lattice points.

First let A consist of the columns of the 2 × (d + 1) matrix

A = (\begin{matrix}d & d − 1 & ⋯ & 1 & 0 \\ 0 & 1 & ⋯ & d − 1 & d. \end{matrix})

The columns give the Laurent monomials defining the rational normal curve C_{d} in Example 0.1. It follows that C_{d} is a projective toric variety. The ideal of C_{d} is the homogeneous ideal given in Example 0.1, and the corresponding affine hyperplane of ℤ^{2} containing A (= the columns of A) consists of all points (a, b) satisfying a + b = d. It is equally easy to see that (1, …, 1) is in the row space of A. In particular, we have

X_{A} = C_{d} and Y_{A} = Ĉ_{d}.

Now let B = {0, 1, …, d − 1, d} ⊆ ℤ. This gives the map
Φ_{B} : ℂ^{∗} ⟶ ℙ^{d}, t ↦ (1, t, …, t^{d−1}, t^{d}).

The resulting projective variety is the rational normal curve, i.e., X_{B} = C_{d}, but the affine variety of B is not the rational normal cone, i.e., Y_{B} ≠ Ĉ_{d}. This can be seen by observing that x_{1}^{2} − x_{2} vanishes at (1, t, …, t^{d−1}, t^{d}) ∈ ℂ^{d+1} for all t ∈ ℂ^{∗}. Thus x_{1}^{2} − x_{2} ∈ I(Y_{B}), so that I(Y_{B}) is not homogeneous. ⋄

Given any A ⊆ M, there is a standard way to modify A so that the conditions of Proposition 1.4 are met: use A × {1} ⊆ M ⊕ ℤ. This lattice corresponds to the torus T_{n} × ℂ^{∗}, and since

(1.4) Φ_{A ×{1}}(t, μ) = (χ^{m_{1}}(t)μ, …, χ^{m_{s}}(t)μ) = μ · (χ^{m_{1}}(t), …, χ^{m_{1}}(t)),

it follows immediately that X_{A ×{1}} = X_{A} ⊆ ℙ^{s−1}. Since A × {1} lies in an affine hyperplane missing the origin, Proposition 1.4 implies that X_{A} has affine cone Y_{A ×{1}} = X̂_{A} . When M = ℤ^{n} and A is represented by the columns of an n × s integer matrix, we obtain A × {1} by adding a row of 1s to the matrix.

The Torus of a Projective Toric Variety. Our next task is to determine the torus of X_{A} . We will do so by identifying the character lattice. This will also tell us the dimension of X_{A} . Given A = {m_{1}, …, m_{s}} ⊆ M, we set

ℤ′A = {∑_{i=1}^{s}a_{i}m_{i} | a_{i} ∈ ℤ, ∑_{i=1}^{s}a_{i} = 0}.

The rank of ℤ′A is the dimension of the smallest affine subspace of M_{ℝ} containing the set A (Exercise 1.3).

Proposition 1.6. Let X_{A} be the projective toric variety of A ⊆ M. Then:

(a) The lattice ℤ′A is the character lattice of torus of X_{A} .

(b) The dimension of X_{A} is the dimension of the smallest affine subspace of M_{ℝ} containing A. In particular,

dim X_{A} = \begin{cases}rank ℤA − 1 & ifY_{A} is the affine cone ofX_{A} \\ rank ℤA & otherwise. \end{cases}

Proof. To prove part (a), let M(T_{X_{A}}) be the character lattice of the torus T_{X_{A}} of X_{A} . By (1.2), we have the commutative diagram

\xymatrix{T_{N} \ar[r] \ar@{->>}[dr] & ℙ^{s−1} \\  & T_{X_{A}} . \ar@{^{(}->}[u]}

which induces the commutative diagram of character lattices

\xymatrix{M & M_{s−1} \ar[l] \ar@{->>}[d] \\  & M(T_{X_{A}}) \ar@{^{(}->}[ul]}

since M_{s−1} = {(a_{1}, …, a_{s}) ∈ ℤ^{s} | ∑_{i=0}^{s} a_{i}m_{i} = 0} is the character lattice of T_{ℙ^{s−1}} by (1.1). The map M_{s−1} → M is induced by the map ℤ^{s} → M that sends e_{i} to m_{i}. Thus ℤ′A is the image of M_{s−1} → M and hence equals M(T_{X_{A}}) by the above diagram.

The first assertion of part (b) follows from part (a) and the observation that rank ℤ′A is the dimension of the smallest affine subpace of M_{ℝ} containing A. Furthermore, if Y_{A} equals the affine cone of X_{A} , then there is u ∈ N with ⟨m_{i}, u⟩ = k > 0 for all i by Proposition 1.4. Thus ⟨∑_{i=1}^{s} a_{i}m_{i}, u⟩ = k(∑_{i=1}^{s} a_{i}), which gives the exact sequence

0 ⟶ ℤ′A ⟶ ℤA ⟶^{⟨ ,u⟩} kℤ ⟶ 0.

Then k > 0 implies rank ℤA − 1 = rank ℤ′A = dim X_{A} . On the other hand, if Y_{A} ≠ X̂_{A} , then the ideal I_{L} is not homogeneous, where L is defined by

0 ⟶ L ⟶ ℤ^{s} ⟶ ℤA ⟶ 0.

by (1.3). Using the usual description of I_{L}, it follows that m · (1, …, 1) ≠ 0 for some m ∈ L. Hence there is ℓ > 0 such that we have an diagram

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & L ∩ M_{s−1} \ar[r] \ar[d] & L \ar[r] \ar[d] & ℓℤ \ar[r] \ar[d] & 0 \\ 0 \ar[r] & M_{s−1} \ar[r] \ar[d] & ℤ^{s} \ar[r] \ar[d] & ℤ \ar[r] \ar[d] & 0 \\ 0 \ar[r] & ℤ′A \ar[r] \ar[d] & ℤA \ar[r] \ar[d] & ℤ/ℓℤ \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

with exact rows and columns. Hence rank ℤA = rank ℤ′A = dim X_{A} . □

Example 1.7. Let A = {e_{1}, e_{2}, e_{1} + 2e_{2}, 2e_{1} + e_{2}} ⊆ ℤ^{2}. One computes that ℤA = ℤ^{2} but ℤ′A = {(a, b) ∈ ℤ^{2} | a + b ≡ 0 mod 2}. Thus ℤ′A has index 2 in ℤA. This means that Y_{A} ≠ X̂_{A} and the map of tori

T_{Y_{A}} ⟶ T_{X_{A}}

is two-to-one, i.e., its kernel has order 2 (Exercise 1.4). ⋄ Affine Pieces of a Projective Toric Variety. So far, our treatment of projective toric varieties has used lattice points and toric ideals. Where are the semigroups? There are actually lots of semigroups, one for each affine piece of X_{A} ⊆ ℙ^{s−1}.

The affine open set U_{i} = ℙ^{s−1} \ V(x_{i}) contains the torus T_{ℙ^{s−1}}. Thus

T_{X_{A}} = X_{A} ∩ T_{ℙ^{s−1}} ⊆ X_{A} ∩ U_{i}.

Since X_{A} is the Zariski closure of T_{X_{A}} in ℙ^{s−1}, it follows that X_{A} ∩ U_{i} is the Zariski closure of T_{X_{A}} in U_{i} ≃ ℂ^{s−1}. Thus X_{A} ∩ U_{i} is an affine toric variety.

Given A = {m_{1}, …, m_{s}} ⊆ M_{ℝ}, the affine semigroup associated to X_{A} ∩U_{i} is easy to determine. Recall that U_{i} ≃ ℂ^{s−1} is given by

(a_{1}, …, a_{s}) ⟼ (a_{1}/a_{i}, …, a_{i−1}/a_{i}, a_{i+1}/a_{i}, …, a_{s}/a_{i}).

Combining this and χ^{m_{j}}/χ^{m_{i}} = χ^{m_{j}−m_{i}} with the map (1.2), we see that X_{A} ∩ U_{i} is the Zariski closure of the image of the map

T_{N} ⟶ ℂ^{s−1}

given by

(1.5) t ⟼ (χ^{m_{1}−m_{i}}(t), …, χ^{m_{i−1}−m_{i}}(t), χ^{m_{i+1}−m_{i}}(t), …, χ^{m_{s}−m_{i}}(t)).

If we set A_{i} = A − m_{i} = {m_{j} − m_{i} | j ≠ i}, it follows that

X_{A} ∩ U_{i} = X_{A_{i}} = Spec(ℂ[S_{i}]),

where S_{i} = ℕA_{i} is the affine semigroup generated by A_{i}. We have thus proved the following result.

Proposition 1.8. Let X_{A} ⊆ ℙ^{s−1} for A = {m_{1}, …, m_{s}} ⊆ M_{ℝ}. Then the affine piece X_{A} ∩ U_{i} is the affine toric variety

X_{A} ∩ U_{i} = X_{A_{i}} = Spec(ℂ[S_{i}])

where A_{i} = A − m_{i} and S_{i} = ℕA_{i}. □

We also note that the results of Chapter 1 imply that the torus of X_{A_{i}} has character lattice ℤA_{i}. Yet the torus is T_{X_{A}}, which has character lattice ℤ′A by Proposition 1.6. These are consistent since ℤA_{i} = ℤ′A for all i.

Besides describing the affine pieces X_{A} ∩ U_{i} of X_{A} ⊂ ℙ^{s−1}, we can also describe how they patch together. In other words, we can give a completely toric description of the inclusions

X_{A} ∩ U_{i} ⊇ X_{A} ∩ U_{i} ∩ U_{j} ⊆ X_{A} ∩ U_{j}

when i ≠ j. For instance, U_{i} ∩ U_{j} consists of those points of X_{A} ∩ U_{i} where x_{j}/x_{i} ≠ 0. In terms of X_{A} ∩ U_{i} = Spec(ℂ[S_{i}]), this means those points where χ^{m_{j}−m_{i}} ≠ 0. Thus

(1.6)  X_{A} ∩ U_{i} ∩ U_{j} = Spec(ℂ[S_{i}])_{χ^{m_{j} −m_{i}}} = Spec(ℂ[S_{i}]_{χ^{m_{j} −m_{i}}} ).

so that if we set m = m_{j} − m_{i}, so that the inclusion X_{A} ∩ U_{i} ∩ U_{j} ⊆ X_{A} ∩ U_{i} can be written

(1.7)  Spec(ℂ[S_{i}])_{χ^{m}} ⊆ Spec(ℂ[S_{i}]).

This looks very similar to the inclusion constructed in (3.4) of Chapter 1 using faces of cones. We will see in §3 that this is no accident.

We next say a few words about how the polytope P = Conv(A) ⊆ M_{ℝ} relates to X_{A} . As we will learn in §2, the dimension of P is the dimension of the smallest affine subspace of M_{ℝ} containing P, which is the same as the smallest affine subspace of M_{ℝ} containing A. It follows from Proposition 1.6 that

dim X_{A} = dim P.

Furthermore, the vertices of P give an especially efficient affine covering of X_{A} .

Proposition 1.9. Given A = {m_{1}, …, m_{s}} ⊆ M, let P = Conv(A) ⊆ M_{ℝ} and set J = {j ∈ {1, …, s} | m_{j} is a vertex ofP }. Then

X_{A} = ⋃_{j∈J} X_{A} ∩ U_{j}.

Proof. We will prove that if i ∈ {1, …, s}, then X_{A} ∩ U_{i} ⊆ X_{A} ∩ U_{j} for some j ∈ J. The discussion of polytopes from §2 below implies that

P ∩ M_{Q} = {_{∑j∈J}r_{j}m_{j} | r_{j} ∈ ℚ_{≥0}, _{∑j∈J}r_{j} = 1}.

Given i ∈ {1, …, s}, we have m_{i} ∈ P ∩ M, so that m_{i} is a convex ℚ-linear combination of the vertices m_{j}. Clearing denominators, we get integers k > 0 and k_{j} ≥ 0 such that

km_{i} = _{∑j∈J}k_{j}m_{j}, _{∑j∈J}k_{j} = k.

Thus ∑_{j∈J} k_{j}(m_{j} − m_{i}) = 0, which implies m_{i} − m_{j} ∈ S_{i} when k_{j} > 0. Fix such a j. Then χ^{m_{j}−m_{i}} ∈ ℂ[S_{i}] is invertible, so ℂ[S_{i}]_{χ^{m_{j} −m_{i}}} = ℂ[S_{i}]. By (1.6), X_{A} ∩ U_{i} ∩ U_{j} = Spec(ℂ[S_{i}]) = X_{A} ∩ U_{i}, giving X_{A} ∩ U_{i} ⊆ X_{A} ∩ U_{j}. □

We conclude with an application to projective normality. An irreducible variety V ⊆ ℙ^{n} is called projectively normal if its affine cone V̂ ⊆ ℂ^{n+1} is normal. A projectively normal variety is always normal (Exercise 1.5). Here is an example to show that the converse can fail.

Example 1.10. Let A ⊆ ℤ^{2} consist of the columns of the matrix

(\begin{matrix}4 & 3 & 1 & 0 \\ 0 & 1 & 3 & 4 \end{matrix}) ,

giving the Laurent monomials s^{4}, s^{3}t, st^{3}, t^{4}. The polytope P = Conv(A) is the line segment connecting (4, 0) and (0, 4), with vertices corresponding to s^{4} and t^{4}. The affine piece of X_{A} corresponding to s^{4} has coordinate ring

ℂ[s^{3}t/s^{4}, st^{3}/s^{4}, t^{4}/s^{4}] = ℂ[t/s, (t/s)^{3}, (t/s)^{4}] = ℂ[t/s],

58 2. Projective Toric Varieties

which is normal since it is a polynomial ring. Similarly, one sees that the coordinate ring corresponding to t^{4} is ℂ[s/t], also normal. These affine pieces cover X_{A} by Proposition 1.9, so that X_{A} is normal.

Since (1, 1, 1, 1) is in the row space of the matrix, Y_{A} is the affine cone of X_{A} by Proposition 1.4. The affine variety Y_{A} is not normal by Example 3.9 of Chapter 1, so that X_{A} is normal but not projectively normal. ⋄

The notion of normality used in this example is a bit ad-hoc since we have not formally defined normality for projective varieties. Once we define normality for abstract varieties in Chapter 3, we will see that Example 1.10 is fully rigorous.

We will say more about projective normality when we explore the connection with polytopes suggested by the above results.

Exercises for § 1.

1.1. Consider the set P_{3} ⊆ ℤ^{3×3} of 3 × 3 permutation matrices defined in Example 1.3.

(a) Prove the claim made in Example 1.3 that three of the permutation matrices sum to the other three and use this to explain why x_{123}x_{231}x_{312} − x_{132}x_{321}x_{213} ∈ I(X_{P_{3}} ).

(b) Show that dim X_{P_{3}} = 4 by computing ℤ′P_{3}.

(c) Conclude that I(X_{P_{3}} ) = ⟨x_{123}x_{231}x_{312} − x_{132}x_{321}x_{213}⟩.

1.2. Let A ⊆ ℤ^{n} consist of the columns of an n × s matrix with integer entries. Prove that the conditions of Proposition 1.4 are equivalent to the assertion that (1, …, 1) ∈ ℤ^{s} lies in the row space of the matrix over ℝ or ℚ.

1.3. Given a finite set A ⊆ M, prove that the rank of ℤ′A equals the dimension of the smallest affine subspace (over ℚ or ℝ) containing A.

1.4. Verify the claims made in Example 1.7. Also compute I(Y_{A}) and check that it is not homogeneous.

1.5. Let V ⊆ ℙ^{n} be projectively normal. Use (0.6) to prove that the affine pieces V ∩ U_{i} of V are normal.

1.6. Fix a finite subset A ⊆ M. Given m ∈ M, let m + A = {m + m′ | m′ ∈ A }. This is the translate of A by m.

(a) Prove that A and its translate m + A give the same projective toric variety, i.e., X_{A} = X_{m+A}.

(b) Give an example to show that the affine toric varieties Y_{A} and Y_{m+A} can differ. Hint: Pick A so that it lies in an affine hyperplane not containing the origin. Then translate A to the origin.

1.7. In Proposition 1.4, give a direct proof that (d) ⇒ (c).

2. Lattice Points and Polytopes

Before we can begin our exploration of the rich connections between toric varieties and polytopes, we first need to study polytopes and their lattice points.

Polytopes. Recall from Chapter 1 that a polytope P ⊆ M_{ℝ} is the convex hull of a finite set S ⊆ M_{ℝ}, i.e., P = Conv(S). Similar to what we did for cones, our discussion of polytopes will omit proofs. Detailed treatments of polytopes can be found in [4, 15].

The dimension of a polytope P ⊆ M_{ℝ} is the dimension of the smallest affine subspace of M_{ℝ} containing P. Given a nonzero vector u in the dual space N_{ℝ} and b ∈ ℝ, we get the affine hyperplane H_{u,b} and closed half-space H_{u,b}^{+} defined by

H_{u,b} = {m ∈ M_{ℝ} | ⟨m, u⟩ = b} and H_{u,b}^{+} = {m ∈ M_{ℝ} | ⟨m, u⟩ ≥ b}.

A subset Q ⊆ P is a face of P if there are u ∈ N_{ℝ} \ {0} and b ∈ ℝ such that

Q = H_{u,b} ∩ P and P ⊆ H_{u,b}^{+}.

We say that H_{u,a} is a supporting affine hyperplane in this situation. Figure 1 shows a polygon with the supporting lines of its faces. The arrows in the figure indicate the vectors u.

Figure 1. A polygon P and four of its supporting lines

We also regard P as a face of itself. Every face of P is again a polytope, and if P = Conv(S) and Q = H_{u,b} ∩ P as above, then Q = Conv(S ∩ H_{u,b}). Faces of P of special interest are facets, edges and vertices, which are faces of dimension dim P − 1, 1 and 0 respectively. Facets will usually be denoted by the letter F.

Here are some properties of faces.

Proposition 2.1. Let P ⊆ M_{ℝ} be a polytope.

(a) P is the convex hull of its vertices.

(b) If P = Conv(S), then every vertex of P lies in S.

(c) If Q is a face of P, then the faces of Q are precisely the faces of P lying in Q.

(d) Every proper face Q is the intersection of the facets F containing Q. □

A polytope P ⊆ M_{ℝ} can also be written as a finite intersection of closed halfspaces. The converse is true provided the intersection is bounded. In other words, if an intersection

P = ⋂_{i=1}^{s} H_{u_{i},b_{i}}^{+}

is bounded, then P is a polytope. Here is a famous example.

Example 2.2. A d × d matrix M ∈ ℝ^{d×d} is doubly-stochastic if it has nonnegative entries and its row and sum columns are all 1. If we regard ℝ^{d×d} as the affine space ℝ^{d^{2}} with coordinates a_{ij}, then the set M_{d} of all doubly-stochastic matrices is defined by the inequalites

a_{ij} ≥ 0 (all i, j)
∑_{i=1}^{d}a_{ij} ≥ 1, ∑_{i=1}^{d}a_{ij} ≤ 1 (all j)
∑_{j=1}^{d}a_{ij} ≥ 1, ∑_{j=1}^{d}a_{ij} ≤ 1 (all i).

(We use two inequalities to get one equality.) These inequalities easily imply that 0 ≤ a_{ij} ≤ 1 for all i, j, so that M_{d} is bounded and hence is a polytope.

Birkhoff and Von Neumann proved independently that the vertices of M_{d} are the d! permutation matrices and that dim M_{d} = (d − 1)^{2}. In the literature, M_{d} has various names, including the Birkhoff polytope and the transportation polytope. See [35, p. 20] for more on this interesting polytope. ⋄

When P has maximal dimension, i.e., dim P = dim M_{ℝ}, its presentation as an intersection of closed half-spaces has an especially nice form because each facet F has a unique supporting affine hyperplane. We write the supporting affine hyperplane and corresponding closed half-space as

H_{F} = {m ∈ M_{ℝ} | ⟨m, u_{F} ⟩ = −a_{F} } and H_{F}^{+} = {m ∈ M_{ℝ} | ⟨m, u_{F} ⟩ ≥ −a_{F} },

where (u_{F} , a_{F}) ∈ N_{ℝ}×ℝ is unique up to multiplication by a positive real number. We call u_{F} an inward-pointing facet normal of the facet F. It follows that

(2.1) P = ⋂_{F facet}H_{F}^{+} = {m ∈ M_{ℝ} | ⟨m, u_{F} ⟩ ≥ −a_{F} for all facets F of P }.

In Figure 1, the supporting lines plus arrows determine the supporting half-planes whose intersection is the polygon P. We write (2.1) with minus signs in order to simplify formulas in later chapters.

Here are some important classes of polytopes.

Definition 2.3. Let P ⊆ M_{ℝ} be a polytope of dimension d.

(a) P is a simplex or d-simplex if it has d + 1 vertices.

(b) P is simplicial if every facet of P is simplicial.

(c) P is simple if every vertex is the intersection of precisely d facets.

Examples include the Platonic solids in ℝ^{3}:

• A tetrahedron is a 3-simplex.

• The octahedron and icosahedron are simplicial since their facets are triangles.

• The cube and dodecahedron are simple since three facets meet at every vertex.

Polytopes P_{1} and P_{2} are combinatorially equivalent if there is a bijection

{faces of P_{1}} ≃ {faces of P_{2}}

that preserves dimensions, inclusions and intersections. For example, simplices of the same dimension are combinatorially equivalent, and in the plane, the same holds for polygons with the same number of vertices.

Sums, Multiples, and Duals. Given a polytope P = Conv(S), its multiple rP = Conv(rS) is again a polytope for any r ≥ 0. If P is defined by the inequalities

⟨m, u_{i}⟩ ≥ a_{i}, 1 ≤ i ≤ s,

then rP is given by

⟨m, u_{i}⟩ ≥ ra_{i}, 1 ≤ i ≤ s.

In particular, when P has maximal dimension, then P and rP have the same inward-pointing facet normals.

The Minkowski sum of subsets A_{1}, A_{2} ⊆ M_{ℝ} is

A_{1} + A_{1} = {a_{1} + a_{2} | a_{1} ∈ A_{1}, a_{2} ∈ A_{2}}.

Given polytopes P_{1} = Conv(S_{1}) and P_{1} = Conv(S_{2}), their Minkowski sum P_{1} + P_{2} = Conv(S_{1} + S_{2}) is again a polytope. We also have the distributive law

rP + sP = (r + s)P.

When P has maximal dimension and 0 is an interior point of P, we define the dual or polar polytope

P^{∘} = {u ∈ N_{ℝ} | ⟨m, u⟩ ≥ −1 for all u ∈ P }.

Figure 2 shows an example of this in the plane.

Figure 2. A polygon P and its dual P^{∘} in the plane

When we write P = {m ∈ M_{ℝ} | ⟨m, u_{F} ⟩ ≥ −a_{F} , F facet} as in (2.1), we have a_{F} > 0 for all F since 0 is in the interior. Then P^{∘} is the convex hull of the vectors (1/a_{F})u_{F} ∈ N_{ℝ} (Exercise 2.1). We also have (P^{∘})^{∘} = P in this situation.

Lattice Polytopes. Now let M and N be dual lattices with associated vector spaces M_{ℝ} and N_{ℝ}. A lattice polytope P ⊆ M_{ℝ} is the convex hull of a finite set S ⊆ M. It follows easily that a polytope in M_{ℝ} is a lattice polytope if and only if its vertices lie in M (Exercise 2.2).

Example 2.4. The standard n-simplex in ℝ^{n} is

Δ_{n} = Conv(0, e_{1}, …, e_{n}).

Another simplex in ℝ^{3} is P = Conv(0, e_{1}, e_{2}, e_{1} + e_{2} + 3e_{3}), shown in Figure 3.

Figure 3. The simplex P = Conv(0, e_{1}, e_{2}, e_{1} + e_{2} + 3e_{3}) ⊆ ℝ^{3}

The lattice polytopes Δ_{3} and P are combinatorially equivalent but will give very different projective toric varieties. ⋄

Example 2.5. The Birkhoff polytope defined in Example 2.2 is a lattice polytope relative to the lattice of integer matrices ℤ^{d×d} since its vertices are the permutation matrices, whose entries are all 0 or 1. ⋄

One can show that faces of lattice polytopes are again lattice polytopes and that Minkowski sums and integer multiples of lattice polytopes are lattice polytopes (Exercise 2.2). Furthermore, every lattice polytope is an intersection of closed half-spaces defined over M, i.e., P = ⋂_{i=1}^{s} H_{u_{i},b_{i}}^{+} where u_{i} ∈ N and b_{i} ∈ ℤ.

When a lattice polytope P has maximal dimension, the facet presentation given in (2.1) has an especially nice form. If F is a facet of P, then the inward-pointing facet normals of F lie on a rational ray in N_{ℝ}. Let u_{F} denote the unique ray generator. The corresponding a_{F} is integral since ⟨m, u_{F} ⟩ = −a_{F} when m is a vertex of F. It follows that

(2.2)  P = {m ∈ M_{ℝ} | ⟨m, u_{F} ⟩ ≥ −a_{F} for all facets F of P }.

is the unique facet presentation of the lattice polytope P.

Example 2.6. Conisder the square P = Conv(±e_{1} ±e_{2}) ⊆ ℝ^{2}. The facet normals of P are ±e_{1} and ±e_{2} and the facet presentation of P is given by the inequalities

⟨m, ±e_{1}⟩ ≥ −1
⟨m, ±e_{2}⟩ ≥ −1.

Since the a_{F} are all equal to 1, it follows that P^{∘} = Conv(±e_{1}, ±e_{2}) is also a lattice polytope. The polytopes P and P^{∘} are pictured in Figure 2.

It is rare that the dual of a lattice polytope is a lattice polytope—this is related to the reflexive polytopes that will be studied later in the book.

Example 2.7. The 3-simplex P = Conv(0, e_{1}, e_{2}, e_{1} + e_{2} + 3e_{3}) ⊆ ℝ^{3} pictured in Example 2.4 has facet presentation

⟨m, e_{3}⟩ ≥ 0
⟨m, 3e_{1} − e_{3}⟩ ≥ 0
⟨m, 3e_{2} − e_{3}⟩ ≥ 0
⟨m, −3e_{1} − 3e_{2} + e_{3}⟩ ≥ −3

(Exercise 2.3). However, if we replace −3 with −1 in the last inequality, we get integer inequalities that define (1/3)P, which is not a lattice polytope. ⋄

The combinatorial type of a polytope is an interesting object of study. This leads to the question “Is every polytope combinatorially equivalent to a lattice polytope?” If given polytope is simplicial, the answer is “yes”—just wiggle the vertices to make them rational and clear denominators to get a lattice polytope. The same argument works for simple polytopes by wiggling the facet normals. This will enable us to prove results about arbitrary simplicial or simple polytopes using toric varieties. But in general, the answer is “no”—there exist polytopes in every dimension ≥ 8 not combinatorially equivalent to any lattice polytope. An example is described in [35, Example 6.21].

Normal Polytopes. The connection between lattice polytopes and toric varieties comes from the lattice points of the polytope. Unfortunately, a lattice polytope might not have enough lattice points. The 3-simplex P from Example 2.7 has only four lattice points (its vertices), which implies that the projective toric variety X_{P ∩ℤ^{3}} is just ℙ^{3} (Exercise 2.3).

We will explore two notions of what it means for a lattice polytope to “have enough lattice points.” Here is the first.

Definition 2.8. A lattice polytope P ⊆ M_{ℝ} is normal if

(kP) ∩ M + (ℓP) ∩ M = ((k + ℓ)P) ∩ M.

for all k, ℓ ∈ ℕ.

The inclusion (kP) ∩ M + (ℓP) ∩ M ⊆ ((k + ℓ)P) ∩ M is automatic. Thus normality means that all lattice points of (k + ℓ)P come from lattice points of kP and ℓP. In particular, a lattice polytope is normal if and only if

\underbrace{P ∩ M + ⋯ + P ∩ M}_{k times} = (kP) ∩ M.

for all integers k ≥ 1. In other words, normality says that P has enough lattice points to generate the lattice points in all integer multiples of P.

Definition 2.9. A simplex P ⊆ M_{ℝ} is basic if P has a vertex m_{0} such that the differences m − m_{0}, for vertices m ≠ m_{0} of P, form a subset of a ℤ-basis of M.

This definition is independent of which vertex m_{0} ∈ P is chosen. The standard simplex Δ_{n} ⊆ ℝ^{n} is basic, and any basic simplex is normal (Exercise 2.4).

Here is another interesting class of normal polytopes.

Proposition 2.10. Every lattice polygon P ⊆ ℝ^{2} is normal.

Proof. In Exercise 2.5 you will show that P is a union of lattice triangles P = ⋃_{i} T_{i} such that each T_{i} has only vertices as lattice points. Such lattice triangles are basic by Exercise 2.6. Take a lattice point p of (k + ℓ)P. This lies in (k + ℓ)T_{i} for some i. The basic triangle T_{i} is normal, so that p lies in

(kT_{i}) ∩ ℤ^{2} + (ℓT_{i}) ∩ ℤ^{2} ⊆ (kP) ∩ ℤ^{2} + (ℓP) ∩ ℤ^{2}.  □

The property “lattice triangles with only vertices as lattice points are basic and hence normal” fails in higher dimensions.

Example 2.11. Let P = Conv(0, e_{1}, e_{2}, e_{1} + e_{2} + 3e_{3}) ⊆ ℝ^{3}. We noted earlier that the only lattice points of P are its vertices. But P is not normal since

e_{1} + e_{2} + e_{3} = \frac{1}{6}(0) + \frac{1}{3}(2e_{1}) + \frac{1}{3}(2e_{2}) + \frac{1}{6}(2e_{1} + 2e_{2} + 6e_{3}) ∈ 2P

is not the sum of lattice points of P. It follows that P is not basic. ⋄

We can also interpret normality in terms of the cone

C_{P} = Cone(P × {1}) ⊆ M_{ℝ} × ℝ

introduced in Figure 3 of Chapter 1. The key feature of this cone is that kP is the “slice” of C_{P} at height k, as illustrated in Figure 4 on the next page. It follows that lattice points m ∈ kP correspond to points (m, k) ∈ C_{P} ∩ (M × ℤ).

Figure 4. The cone C_{P} sliced at heights 1 and 2

In Exercise 2.7 you will show that the semigroup C_{P} ∩ (M × ℤ) of lattice points in C_{P} relates to normality as follows.

Lemma 2.12. The lattice polytope P ⊆ M_{ℝ} is normal if and only if (P ∩M)×{1} generates the semigroup C_{P} ∩ (M × ℤ). □

Here is an important result on normality.

Theorem 2.13. Let P ⊆ M_{ℝ} be a lattice polytope. Then there is an integer k_{0} > 0 such that kP is normal for all k ≥ k_{0}.

Proof. Let H = {(m_{i}, h_{i})}, i = 1, …, s, be the Hilbert basis of C_{P} ∩ (M × ℤ). Note that m_{i} ∈ h_{i}P. In Exercise 2.8 you will adapt the proof of Gordan’s Lemma (Proposition 2.15 from Chapter 1) to show that if H is the Hilbert basis of the semigroup of lattice points in a strongly convex cone Cone(A), then the lattice points in the cone can be written as the union

(2.3)  ℕA ∪_{⋃m∈H} (m + ℕA ).

Applied to A = (P ∩ M) × {1} ⊆ M × ℤ, this implies that

(2.4)  C_{P} ∩ (M × ℤ) = S ∪ ⋃_{i=1}^{s}((m_{i}, h_{i}) + S),

where S = ℕ((P ∩ M) × {1}).

Now let k_{0} = max{h_{i} | i = 1, …, s} and take k ≥ k_{0}. We will prove that kP is normal by showing that

(kP) ∩ M + (dkP) ∩ M = ((d + 1)kP) ∩ M

for all d ∈ ℕ. It suffices to prove ⊇, so let m ∈ ((d + 1)kP) ∩ M. Then (m, (d + 1)k) ∈ C_{P} ∩ (M × ℤ), which means that (m, (d + 1)k) lies in the union (2.4). If (m, (d + 1)k) ∈ S, then m is a sum of (d + 1)k points from P ∩ M. Writing this as a sum of k points plus a sum of dk points shows that m ∈ (kP) ∩ M + (dkP) ∩ M. Also, if (m, (d + 1)k) ∈ (m_{i}, h_{i}) + S, then

m = m_{i} + (sum of (d + 1)k−h_{i} points of P ∩ M).

Since k ≥ h_{i}, we can write this as

m = (m_{i} + sum of k−h_{i} points of P ∩ M) + (sum of dk points of P ∩ M).

Since m_{i} ∈ h_{i}P, it follows that m ∈ (kP) ∩ M + (dkP) ∩ M. □

Here is an example of this result.

Example 2.14. Consider P = Conv(0, e_{1}, e_{2}, e_{1} + e_{2} + 3e_{3}) from Example 2.11. The Hilbert basis of C_{P} ∩ (M × ℤ) consists of the six points

(0, 1), (e_{1}, 1), (e_{2}, 1), (e_{1} + e_{2} + 3e_{3}, 1), (e_{1} + e_{2} + e_{3}, 2)(e_{1} + e_{2} + 2e_{3}, 2)

(Exercise 2.3). It follows from the proof of Theorem 2.13 that 2P is normal.

In Chapter 6 we will sharpen Theorem 2.13 by showing that if P is an ndimensional lattice polytope, then kP is normal for all k ≥ n − 1. This was first proved in [6] and explains why 2P is normal in Example 2.14. Also, in Chapter 7 we will show that the number of lattice points in nP is given by a polynomial in n called the Ehrhart polynomial.

Very Ample Polytopes. Here is a slightly different notion of what it means for a polytope to have enough lattice points.

Definition 2.15. A lattice polytope P ⊆ M_{ℝ} is very ample if for every vertex m ∈ P, the semigroup S_{P,m} = ℕ(P ∩ M − m) generated by P ∩ M − m = {m′ − m | m′ ∈ P ∩ M} is saturated in M.

This definition relates to normal polytopes as follows.

Proposition 2.16. A normal lattice polytope P is very ample.

Proof. Fix a vertex m_{0} ∈ P and take m ∈ M such that km ∈ S_{P,m_{0}} for some integer k ≥ 1. To prove that m ∈ S_{P,m_{0}}, write km ∈ S_{P,m_{0}} as

km = _{∑m_{′}∈P ∩M}a_{m′}(m′ − m_{0}), a_{m′} ∈ ℕ.

Pick d ∈ ℕ so that kd ≥ ∑_{m′∈P ∩M} a_{m′}. Then

km + kdm_{0} = _{∑m_{′}∈P ∩M}a_{m′}m′ + (kd −_{∑m_{′}∈P ∩M}a_{m′})m_{0} ∈ kdP.

Dividing by k gives m + dm_{0} ∈ dP, which by normality implies that

m + dm_{0} = ∑_{i=1}^{d} m_{i}, m_{i} ∈ P ∩ M for all i.

We conclude that m = ∑_{i=1}^{d}(m_{i} − m_{0}) ∈ S_{P,m_{0}}, as desired. □

Combining this with Proposition 2.10 and Theorem 2.13 gives the following.

Corollary 2.17. Let P ⊆ M_{ℝ} be a lattice polytope.

(a) If dim P = 2, then P is very ample.

(b) There is an integer k_{0} ≥ 1 such that kP is very ample for all k ≥ k_{0}. □

We will see in the next section that very ampleness is precisely the property we need in order to define the toric variety of a lattice polytope. Also, the sharpening of Theorem 2.13 mentioned earlier will enable us to take k_{0} = dim P −1 in part (b) of the corollary.

The following example taken from [5, Example 5.1] shows that very ample polytopes need not be normal, i.e., the converse of Proposition 2.16 is false.

Example 2.18. Given 1 ≤ i < j < k ≤ 6, let [ijk] denote the vector in ℤ^{6} with 1 in positions i, j, k and 0 elsewhere. Thus [123] = (1, 1, 1, 0, 0, 0). Then let

A = {[123], [124], [135], [146], [156], [236], [245], [256], [345], [346]} ⊆ ℤ^{6}.

The lattice polytope P = Conv(A) lies in the affine hyperplane of ℝ^{6} where the coordinates sum to 3. As explained in [5], this configuration can be interpreted in terms of a triangulation of the real projective plane.

The points of A of P are the only lattice points of P (Exercise 2.9), so that A is the set of vertices of P. Number the points of A as m_{1}, …, m_{10}. Then

(1, 1, 1, 1, 1, 1) = \frac{1}{5} ∑_{i=1}^{10} m_{i} = ∑_{i=1}^{10} \frac{1}{10}(2m_{i})

shows that v = (1, 1, 1, 1, 1, 1) ∈ 2P. Since v is not a sum of lattice points of P (when [ijk] ∈ A, the vector with 1s in the complementary positions is not in A), we conclude that P is not normal.

Showing that P is very ample takes more work. The first step is to prove that A × {1} ∪ {(v, 2)} ⊆ ℝ^{6} × ℝ is a Hilbert basis of the semigroup C_{P} ∩ ℤ^{7}, where C_{P} ⊆ ℝ^{6} × ℝ = ℝ^{7} is the cone over P × {1}. We used the software 4ti2 [18].

Now fix i and let S_{P,m_{i}} be the semigroup generated by the m_{j} − m_{i}. Take m ∈ ℤ^{6} such that km ∈ S_{P,m_{i}}. As in the proof of Proposition 2.16, this implies that m + dm_{i} ∈ dP for some d ∈ ℕ. Thus (m + dm_{i}, d) ∈ C_{P} ∩ ℤ^{7}. Expressing this in terms of the above Hilbert basis easily implies that

m = a(v − 2m_{i}) + ∑_{j=1}^{10} a_{j}(m_{j} − m_{i}), a, a_{j} ∈ ℕ.

If we can show that v − 2m_{i} ∈ S_{P,m_{i}}, then m ∈ S_{P,m_{i}} follows immediately and proves that S_{P,m_{i}} is saturated. When i = 1, one can check that

v + [123] = [124] + [135] + [236],

which implies that

v − 2m_{1} = (m_{2} − m_{1}) + (m_{3} − m_{1}) + (m_{6} − m_{1}) ∈ S_{P,m_{1}}. One obtains similar formulas for i = 2, …, 10 (Exercise 2.9), which completes the proof that P is very ample.

The polytope P has further interesting properties. For example, up to affine equivalence, P can be described as the convex hull of the 10 points in ℤ^{5} given by

(0, 0, 0, 0, 0), (0, 0, 0, 0, 1), (0, 0, 1, 1, 0), (0, 1, 0, 1, 1), (0, 1, 1, 1, 0)

(1, 0, 1, 0, 1), (1, 0, 1, 1, 1), (1, 1, 0, 0, 0), (1, 1, 0, 1, 1), (1, 1, 1, 0, 0).

Of all 5-dimensional polytopes whose vertices lie in {0, 1}^{5}, this polytope has the most edges, namely 45 (see [1]). Since it has 10 vertices and 45 = \binom{10}{2}, every pair of distinct vertices is joined by an edge. Such polytopes are 2-neighborly. ⋄

Exercises for § 2.

2.1. Let P ⊆ M_{ℝ} be a polytope of maximal dimension with the origin as an interior point.

(a) Write P = {m ∈ M_{ℝ} | ⟨m, u_{F} ⟩ ≥ −a_{F} for all facets F }. Prove that a_{F} > 0 for all F and that P^{∘} = Conv((1/a_{F})u_{F} | F a facet).

(b) Prove that the dual of a simplicial polytope is simple and vice versa.

(c) Prove that (rP)^{∘} = (1/r)P^{∘} for all r > 0.

(d) Use part (c) to construct an example of a lattice polytope whose dual is not a lattice polytope.

2.2. Let P ⊆ M_{ℝ} be a polytope.

(a) Prove that P is a lattice polytope if and only if the vertices of P lie in M

(b) Prove that P is a lattice polytope if and only if P is the convex hull of its lattice points, i.e., P = Conv(P ∩ M).

(c) Prove that every face of a lattice polytope is a lattice polytope.

(d) Prove that Minkowski sums and integer multiples of lattice polytopes are again lattice polytopes.

2.3. Let P = Conv(0, e_{1}, e_{2}, e_{1} +e_{2} +3e_{3}) ⊆ ℝ^{3} be the simplex studied in Examples 2.4, 2.7, 2.11 and 2.14.

(a) Verify the facet presentation of P given in Example 2.7.

(b) Show that the only lattice points of P are its vertices.

3. Polytopes and Projective Toric Varieties 69

(c) Show that the toric variety X_{P ∩ℤ^{3}} is ℙ^{3}.

(d) Show that the vectors given in Example 2.14 form the Hilbert basis of C_{P} ∩ (M × ℤ).

2.4. Recall the definition of basic simplex given in Definition 2.9.

(a) Show that if a simplex satisfies Definition 2.9 for one vertex, then it satisfies the definition for all vertices.

(b) Show that the standard simplex Δ_{n} is basic.

(c) Prove that a basic simplex is normal.

2.5. Let P be a lattice polygon. Prove that P can be written as a union of lattice triangles such that each lattice triangle has only vertices as lattice points. Hint: First use induction on the number of vertices to reduce to the case when P is a lattice triangle. Then use induction on the number of lattice points. This will construct a triangulation of P.

2.6. Prove that a lattice triangle with only vertices as lattice points is basic. Hint: Translate one vertex to the origin and consider the cone generated by the other two vertices.

2.7. Prove Lemma 2.12.

2.8. Prove (2.3).

2.9. Consider the polytope P = Conv(A) from Example 2.18.

(a) Prove that A is the set of lattice points of P.

(b) Complete the proof begun in the text that P is very ample.

2.10. Prove that every proper face of a simplicial polytope is a simplex.

2.11. In Corollary 2.17 we proved that kP is very ample for k sufficiently large using Theorem 2.13 and Proposition 2.16. Give a direct proof of this result. Hint: A vertex m ∈ P gives the cone C_{P,m} generated by the semigroup S_{P,m} defined in Definition 2.15. The cone C_{P,m} is strongly convex since m is a vertex and hence C_{P,m} ∩ M has a Hilbert basis. Furthermore, C_{P,m} = C_{kP,km} for all k ∈ ℕ. Now argue that when k is large, (kP) ∩ M − km contains the Hilbert basis of C_{P,m} ∩ M. A picture will help.

2.12. Fix an integer a ≥ 1 and consider the 3-simplex P = Conv(0, ae_{1}, ae_{2}, e_{3}) ⊆ ℝ^{3}.

(a) Work out the facet presentation of P and verify that the facet normals can be labeled so that u_{0} + u_{1} + u_{2} + au_{3} = 0.

(b) Show that P is normal. Hint: Show that P ∩ ℤ^{3} + (kP) ∩ ℤ^{3} = ((k + 1)P) ∩ ℤ^{3}. We will see later that the toric variety of P is the weighted projective space ℙ(1, 1, 1, a).

3. Polytopes and Projective Toric Varieties

Our next task is to define the toric variety of a lattice polytope. As noted in §2, we need to make sure that the polytope has enough lattice points. Hence we begin with very ample polytopes. Strongly convex rational polyhedral cones will play an important role in our development.

The Very Ample Case. Let P ⊆ M_{ℝ} be a very ample polytope relative to the lattice M, and assume that dim P = dim M_{ℝ} = rank M. We define

X_{P} = X_{P ∩M}.

To see what this means concretely, write P ∩ M = {m_{1}, …, m_{s}}. Then X_{P} is the Zariski closure of the image of the map T_{N} → ℙ^{s−1} given by

t ⟼ (χ^{m_{1}}(t), …, χ^{m_{s}}(t)) ∈ ℙ^{s−1}.

Fix homogeneous coordinates x_{1}, …, x_{s} for ℙ^{s−1}.

We examine the structure of X_{P} ⊆ ℙ^{s−1} using Propositions 1.8 and 1.9. For each m_{i} ∈ P ∩ M consider the semigroup

S_{i} = ℕ(P ∩ M − m_{i})

generated by m_{j} − m_{i} for m_{j} ∈ P ∩ M. In ℙ^{s−1} we have the affine open subset U_{i} ≃ ℂ^{s−1} consisting of those points where x_{i} ≠ 0. Proposition 1.8 showed that the affine open piece X_{P} ∩ U_{i} of X_{P} is the affine toric variety

X_{P} ∩ U_{i} ≃ Spec(ℂ[S_{i}]),

and Proposition 1.9 showed that

X_{P} = ⋃_{m_{i} vertex of P} X_{P} ∩ U_{i}.

Here is our first major result about X_{P}.

Theorem 3.1. Let X_{P} be the projective toric variety of the very ample polytope P ⊆ M_{ℝ} ≃ ℝ^{n}, and assume that dim P = n.

(a) For each vertex m_{i} ∈ P ∩M, the affine piece X_{P} ∩U_{i} is the affine toric variety

X_{P} ∩ U_{i} = U_{σ_{i}} = Spec(ℂ[σ_{i}^{∨} ∩ M])

where σ_{i} ⊆ N_{ℝ} is the strongly convex rational polyhedral cone dual to the cone Cone(P ∩ M − m_{i}) ⊆ M_{ℝ}. Furthermore, dim σ_{i} = n.

(b) The torus of X_{P} has character lattice M and hence is the torus T_{N}.

Proof. Let C_{i} = Cone(P ∩ M − m_{i}). Since m_{i} is a vertex, it has a supporting hyperplane H_{u,a} such that P ⊆ H_{u,a}^{+} and P ∩ H_{u,a} = {m_{i}}. It follows that H_{u,0} is a supporting hyperplane of 0 ∈ C_{i} (Exericse 3.1), so that C_{i} is strongly convex. It is also easy to see that dim C_{i} = dim P (Exericse 3.1). It follows that C_{i} and σ_{i} = C_{i}^{∨} are strongly convex rational polyhedral cones of dimension n.

We have S_{i} ⊆ C_{i} ∩ M = σ_{i}^{∨} ∩ M. By hypothesis, P is very ample, which means that S_{i} ⊆ M is saturated. Since S_{i} and C_{i} = σ_{i}^{∨} are both generated by P ∩ M − m_{i}, part (a) of Exercise 3.4 of Chapter 1 implies S_{i} = σ_{i}^{∨} ∩ M. (This exercise was part of the proof of the characterization of normal affine toric vareities given in Theorem 3.5 of Chapter 1.) Part (a) of the theorem follows immediately.

For part (b), Theorem 2.16 implies that T_{N} is the torus of U_{σ_{i}} since σ_{i} is strongly convex. Then T_{N} ⊆ U_{σ_{i}} = X_{P} ∩ U_{i} ⊆ X_{P} shows that T_{N} is also the torus of X_{P}. □

The affine pieces X_{P} ∩ U_{i} and X_{P} ∩ U_{j} intersect in X_{P} ∩ U_{i} ∩ U_{j}. In order to describe this intersection carefully, we need to study how the cones σ_{i} and σ_{j} fit together in N_{ℝ}. This leads to our next topic.

The Normal Fan. The cones σ_{i} ⊆ N_{ℝ} appearing in Theorem 3.1 fit together in a remarkably nice way, giving a structure called the normal fan of P.

Let P ⊆ M_{ℝ} ≃ ℝ^{n} be a lattice polytope of dimension n, not necessarily very ample. Faces, facets and vertices of P will be denoted by Q, F and v respectively. Hence we write the facet presentation of P as

(3.1)  P = {m ∈ M_{ℝ} | ⟨m, u_{F} ⟩ ≥ −a_{F} for all F }.

A vertex v ∈ P gives cones

C_{v} = Cone(P ∩ M − v) ⊆ M_{ℝ} and σ_{v} = C_{v}^{∨} ⊆ N_{ℝ}.

(When v = m_{i}, these are the cones C_{i} and σ_{i} studied above.) Faces Q ⊆ P containing v correspond bijectively to faces Q ⊆ C_{v} via the maps

(3.2)  \begin{matrix}Q & ⟼ Q = Cone(Q ∩ M − v) \\ Q & ⟼ Q = (Q + v) ∩ P \end{matrix}

which are inverses of each other. These maps preserve dimensions, inclusions, and intersections (Exercise 3.2), as illustrated in Figure 5.

Figure 5. The cone C_{v} of a vertex v ∈ P

In particular, all facets of C_{v} come from facets of P containing v, so that

C_{v} = {m ∈ M_{ℝ} | ⟨m, u_{F} ⟩ ≥ 0 for all F containing v}.

By the duality results of Chapter 1, it follows that the dual cone σ_{v} is given by

σ_{v} = Cone(u_{F} | F contains v).

This construction generalizes to arbitrary faces Q ⊆ P by setting

σ_{Q} = Cone(u_{F} | F contains Q).

In particular, σ_{F} is the ray generated by u_{F}, and σ_{P} = {0} since the {0} is the cone generated by the empty set.

The following easy lemma will be useful.

Lemma 3.2. Let Q be a face of P and let H_{u,b} be a supporting affine hyperplane of P. Then u ∈ σ_{Q} if and only if Q ⊆ H_{u,b} ∩ P.

Proof. First suppose that u ∈ σ_{Q} and write u = ∑_{Q⊆F} λ_{F} u_{F}, λ_{F} ≥ 0. Then setting b = − ∑_{Q⊆F} λ_{F} a_{F} easily implies that P ⊆ H_{u,b}^{+} and Q ⊆ H_{u,b} ∩ P. Recall that the integers a_{F} come from the facet presentation (3.1).

Going the other way, suppose that Q ⊆ H_{u,b} ∩ P. Take a vertex v ∈ Q. Then P ⊆ H_{u,b}^{+} and v ∈ H_{u,b} imply that C_{v} ⊆ H_{u,0}^{+}. Hence u ∈ C_{v}^{∨} = σ_{v}, so that

u = _{∑v∈F} λ_{F} u_{F} , λ_{F} ≥ 0.

Let F_{0} be a facet of P containing v but not Q, and pick p ∈ Q with p ∉ F_{0}. Then p, v ∈ Q ⊆ H_{u,b} imply that

b = ⟨p, u⟩ = _{∑v∈F} λ_{F} ⟨p, u_{F} ⟩
b = ⟨v, u⟩ = _{∑v∈F} λ_{F} ⟨v, u_{F} ⟩ = −_{∑v∈F} λ_{F} a_{F} .

where the last equality uses ⟨v, u_{F} ⟩ = −a_{F} for v ∈ F. These equations imply

∑v∈F λ_{F} ⟨p, u_{F} ⟩ = −_{∑v∈F} λ_{F} a_{F} .

However, p ∉ F_{0} gives ⟨p, u_{F_{0}}⟩ > −a_{F_{0}}, and since ⟨p, u_{F} ⟩ ≥ −a_{F} for all F, it follows that λ_{F_{0}} = 0 whenever Q ⊈ F_{0}. This gives u ∈ σ_{Q} and completes the proof of the lemma. □

Corollary 3.3. If Q is a face of F, then u_{F} ∈ σ_{Q} if and only if Q ⊆ F.

Proof. One direction is obvious by the definition of σ_{Q}, and the other direction follows from Lemma 3.2 since H_{u_{F} ,−a_{F}} is a supporting affine hyperplane of P with H_{u_{F} ,−a_{F}} ∩ P = F. □

We are now ready to explore the combinatorial and geometric properties of the cones σ_{Q}. We begin with some combinatorial properties.

Proposition 3.4. Let Q and Q′ be faces of a lattice polytope P ⊆ M_{ℝ} ≃ ℝ^{n} of dimension n. Then:

(a) Q ⊆ Q′ if and only if σ_{Q′} ⊆ σ_{Q}.

(b) If Q ⊆ Q′, then σ_{Q′} is a face of σ_{Q}, and all faces of σ_{Q} are of this form.

(c) σ_{Q} ∩ σ_{Q′} = σ_{Q′′}, where Q′′ is the smallest face of P containing Q and Q′.

Proof. To prove part (a), note that if Q ⊆ Q′, then any facet containing Q′ also contains Q, which implies σ_{Q′} ⊆ σ_{Q}. The other direction follows easily from Corollary 3.3 since every face is the intersection of the facets containing it by Proposition 2.1.

For part (b), fix a vertex v ∈ Q and note that by (3.2), Q determines a face Q of C_{v}. Using the duality of Proposition 2.9 of Chapter 1, Q gives the dual face

Q^{∗} = C_{v}^{∨} ∩ Q^{⊥} = σ_{v} ∩ Q^{⊥}

of the cone σ_{v}. Then using σ_{v} = Cone(u_{F} | v ∈ F) and Q ⊆ C_{v} = σ_{v}^{∨}, one obtains

Q^{∗} = Cone(u_{F} | v ∈ F, Q ⊆ H_{u_{F} ,0}).

Since v ∈ Q, the inclusion Q ⊆ H_{u_{F} ,0} is equivalent to Q ⊆ H_{u_{F} ,−a_{F}}, which in turn is equivalent to Q ⊆ F. It follows that

(3.3)  Q^{∗} = Cone(u_{F} | Q ⊆ F) = σ_{Q},

so that σ_{Q} is a face of σ_{v}, and all faces of σ_{v} arise in this way.

In particular, Q ⊆ Q′ means that σ_{Q′} is also a face of σ_{v}, and since σ_{Q′} ⊆ σ_{Q} by part (a), we see that σ_{Q′} a face of σ_{Q}. Furthermore, every face of σ_{Q} is a face of σ_{v} by Proposition 2.6 of Chapter 1 and hence is of the form σ_{Q′} for some face Q′. Using part (a) again, we see that Q ⊆ Q′, and part (b) follows.

For part (c), let Q′′ be the smallest face of P containing Q and Q′. This exists because a face is the intersection of the facets containing it, so that Q′′ is the intersection of all facets containing Q and Q′ (if there are no such facets, then Q′′ = P). By part (b) σ_{Q′′} is a facet of both σ_{Q} and σ_{Q′}. Thus σ_{Q′′} ⊆ σ_{Q} ∩ σ_{Q′}.

It remains to prove the opposite inclusion. If σ_{Q} ∩ σ_{Q′} = {0} = σ_{P}, then Q′′ = P and we are done. If σ_{Q} ∩ σ_{Q′} ≠ {0}, any nonzero u in the intersection lies in both σ_{Q} and σ_{Q′}. The proof of Proposition 3.6 given below will show that H_{u,b} is a supporting affine hyperplane of P for some b ∈ ℝ. By Lemma 3.2, u ∈ σ_{Q} and u ∈ σ_{Q′} imply that Q and Q′ lie in H_{u,b} ∩ P. The latter is a face of P containing Q and Q′, so that Q′′ ⊆ H_{u,b} ∩ P since Q′′ is the smallest such face. Applying Lemma 3.2 again, we see that u ∈ σ_{Q′′}. □

Proposition 3.4 shows that there is a bijective correspondence between faces and cones. The proposition also implies that the cones fit together as follows.

Corollary 3.5. Let P ⊆ M_{ℝ} ≃ ℝ^{n} be a lattice polytope of dimension n and set Σ_{P} = {σ_{Q} | Q is a face of Q}. Then:

(a) For all σ_{Q} ∈ Σ_{P}, each face of σ_{Q} is also in Σ_{P}.

(b) The intersection σ_{Q} ∩ σ_{Q′} of any two cones in Σ_{P} is a face of each.

In Chapter 3, a collection of strongly convex rational polyhedral cones satisfying conditions (a) and (b) of Corollary 3.5 will be called a fan. Since the cones in the above fan Σ_{P} are built from the inward-pointing normal vectors u_{F}, we call Σ_{P} the normal fan or inner normal fan of P.

Before we draw pictures of normal fans, we mention some of the geometric properties of the cones σ_{Q}.

Proposition 3.6. Let P ⊆ M_{ℝ} ≃ ℝ^{n} be a lattice polytope of dimension n and consider the cones σ_{Q} in the normal fan Σ_{P} of P. Then:

(a) dim Q + dim σ_{Q} = n for all faces Q of P.

(b) N_{ℝ} = ⋃_{v} σ_{v} = ⋃_{Q face of P} σ_{Q}.

Proof. Let Q be a face of P and take a vertex v of Q. By (3.2) this gives a face Q of the cone C_{v}, which has a dual face Q^{∗} of the dual cone C_{v}^{∨} = σ_{v}. Since Q^{∗} = σ_{Q} by (3.3), we have

dim Q + dim σ_{Q} = dim Q + dim Q^{∗} = n,

where the first equality uses Exercise 3.2 and the second follows from Proposition 2.9 of Chapter 1. This proves part (a). For part (b), let u ∈ N_{ℝ} be nonzero and set b = min{⟨v, u⟩ | v vertex of P }. Then P ⊆ H_{u,b}^{+} and v ∈ H_{u,b} for at least one vertex of P, so that u ∈ σ_{v} by Lemma 3.2. To prove the final equality of part (b), note that every face Q contains a vertex v, so that σ_{Q} is a face of σ_{v} by Proposition 3.4. □

In the terminology of Chapter 3, a fan satisfying the condition of part (b) of Proposition 3.6 is called complete. Thus the normal fan of a lattice polytope is always complete.

Example 3.7. Figure 6 on the next page shows a lattice hexagon P in the plane together with its normal fan. The vertices of P are labeled v_{1}, …, v_{6}, with corresponding cone σ_{1}, …, σ_{6} in the normal fan. In the figure, P is shown on the right, and at each vertex v_{i}, we have drawn the normal vectors of the facets containing v_{i} and shaded the cone σ_{i} they generate. On the right, these cones are assembled at the origin to give the normal fan.

Notice how one can read off the structure of P from the normal fan. For example, two cones σ_{i} and σ_{j} share a ray in Σ_{P} if and only if the vertices v_{i} and v_{j} lie on an edge of P. ⋄

Here is a similar example that illustrates an important property of normal fans.

Figure 6. A lattice hexagon P and its normal fan Σ_{P}

Example 3.8. The 2-simplex Δ_{2} ⊆ ℝ^{2} has vertices 0, e_{1}, e_{2}. Let P = kΔ_{2} for some positive integer k. Figure 7 shows P and its normal fan Σ_{P}. At each vertex

Figure 7. The triangle P = kΔ_{2} ⊆ ℝ^{2} and its normal fan Σ_{P}

v_{i} of P, we have drawn the normal vectors of the facets containing v_{i} and shaded the cone σ_{i} they generate. The reassembled cones appear on the left as Σ_{P}. ⋄

An important observation is that the normal fan in Example 3.8 doesn’t depend on the integer k. In general, multiplying a polytope by a positive integer has no effect on its normal fan, and the same is true for translations by lattice points. We record these properties in the following proposition (Exercise 3.3).

Proposition 3.9. Let P ⊆ M_{ℝ} ≃ ℝ^{n} be an n-dimensional lattice polytope. Then for any lattice point m ∈ M and any integer k ≥ 1, the polytopes m + P and kP have the same normal fan as P. □

Here is an example of the normal fan of a polytope in ℝ^{3}.

Example 3.10. Consider the cube P ⊆ ℝ^{3} with vertices (±1, ±1, ±1). The facet normals are ±e_{1}, ±e_{2}, ±e_{3}, and the facet presentation of P is

⟨m, ±e_{i}⟩ ≥ −1.

The origin is an interior point of P. By Exercise 2.1, the facet normals are the vertices of the dual polytope P^{∘}, the octahedron pictured in Figure 8.

Figure 8. A cube P ⊆ ℝ^{3} and its dual octahedron P^{∘}

However, the facet normals also give the normal fan of P, and one can check that in the above figure, the maximal cones of the normal fan are the octants of ℝ^{3}, which are just the cones over the facets of the dual polytope P^{∘}. ⋄

As noted earlier, it is rare that both P and P^{∘} are lattice polytopes. However, whenever P ⊆ M_{ℝ} is a lattice polytope containing 0 as an interior point, it is still true that maximal cones of the normal fan Σ_{P} are the cones over the facets of P^{∘} ⊆ N_{ℝ} (Exercise 3.4).

The special behavior of the polytopes P and P^{∘} discussed in Examples 2.6 and 3.10 leads to the following definition.

Definition 3.11. An n-dimensional lattice polytope P ⊆ M_{ℝ} ≃ ℝ^{n} is reflexive if its facet presentation is

P = {m ∈ M_{ℝ} | ⟨m, u_{F} ⟩ ≥ −1 for all facets F }.

If P is reflexive, then 0 is a lattice point of P and and is in fact the only interior lattice point of P (Exercise 3.5). Since a_{F} = 1 for all F, Exercise 2.1 implies that

P^{∘} = Conv(u_{F} | F facet of P).

Thus P^{∘} is a lattice polytope and is in fact reflexive (Exercise 3.5).

We will see later that reflexive polytopes lead to some very interesting toric varieties that are important for mirror symmetry.

The General Case. Earlier in the section, we defined the toric variety X_{P} of a very ample polytope and studied the affine pieces of X_{P}. This development leaves two items of unfinished business:

• What is the interesection two of the affine pieces described in Theorem 3.1?

• How do we define the projective toric variety X_{P} when P is not very ample? As we will see, these questions are linked in a very satisfying manner.

To study the first question, let P ⊆ M_{ℝ} ≃ ℝ^{n} be an n-dimensional very ample polytope and set s = |P ∩ M|. Then

X_{P} = X_{P ∩M} ⊆ ℙ^{s−1}.

Label the homogeneous coordinates of ℙ^{s−1} as x_{m} for m ∈ P ∩M. If v is a vertex of P, then U_{v} ⊆ ℙ^{s−1} is the affine open subset where x_{v} ≠ 0, and Theorem 3.1 tells us that

X_{P} ∩ U_{v} = U_{σ_{v}} = Spec(ℂ[σ_{v}^{∨} ∩ M]).

Thus X_{P} ∩ U_{v} is the affine toric variety of the cone σ_{v} in the normal fan Σ_{P} of P.

Proposition 3.12. Let P ⊆ M_{ℝ} ≃ ℝ^{n} be very ample of dimension n. If v ≠ w are vertices of X_{P} and Q is the smallest face of P containing v and w, then

X_{P} ∩ U_{v} ∩ U_{w} = U_{σ_{Q}} = Spec(ℂ[σ_{Q}^{∨} ∩ M])

and the inclusions

X_{P} ∩ U_{v} ⊇ X_{P} ∩ U_{v} ∩ U_{w} ⊆ X_{P} ∩ U_{w}

can be written

(3.4)  U_{σ_{v}} ⊇ (U_{σ_{v}})_{χ^{w−v}} = U_{σ_{Q}} = (U_{σ_{w}})_{χ^{v−w}} ⊆ U_{σ_{w}}.

Proof. We analyzed the intersection of affine pieces of X_{P ∩M} in §1. Translated to the notation being used here, (1.6) and (1.7) imply that

X_{P} ∩ U_{v} ∩ U_{w} = (U_{σ_{v}})_{χ^{w−v}} = (U_{σ_{w}})_{χ^{v−w}}.

Thus all we need to show is that

(U_{σ_{v}})_{χ^{w−v}} = U_{σ_{Q}}.

However, we have w − v ∈ C_{v} = σ_{v}^{∨}, so that τ = H_{w−v} ∩ σ_{v} is a face of σ_{v}. In this situation, Proposition 3.16 and equation (3.4) of Chapter 1 imply that

(U_{σ_{v}})_{χ^{w−v}} = U_{τ} .

Thus the proposition will follow once we prove τ = σ_{Q}, i.e., H_{w−v} ∩ σ_{v} = σ_{Q}. Since σ_{Q} = σ_{v} ∩ σ_{w} by Proposition 3.4, it suffices to prove that

H_{w−v} ∩ σ_{v} = σ_{v} ∩ σ_{w}.

Let u ∈ H_{w−v} ∩ σ_{v}. If u ≠ 0, there is b ∈ ℝ such H_{u,b} is a supporting affine hyperplane of P. Then u ∈ σ_{v} implies v ∈ H_{u,b} by Lemma 3.2, so that w ∈ H_{u,b} since u ∈ H_{w−v}. Applying Lemma 3.2 again, we get u ∈ σ_{w}. Going the other way, let u ∈ σ_{v} ∩ σ_{w}. If u ≠ 0, pick b ∈ ℝ as above. Then u ∈ σ_{v} ∩ σ_{w} and Lemma 3.2 imply that v, w ∈ H_{u,b}, from which u ∈ H_{w−v} follows easily. This completes the proof. □

This proposition and Theorem 3.1 have the remarkable result that the normal fan Σ_{P} completely determines the internal structure of X_{P}: we build X_{P} from local pieces given by the affine toric varieties U_{σ_{v}}, glued together via (3.4). We don’t need the ambient projective space ℙ^{s−1} for any of this—everything we need to know is contained in the normal fan.

As a consequence, we can now give the general definition of the toric variety of a polytope.

Definition 3.13. Let P ⊆ M_{ℝ} ≃ ℝ^{n} be an n-dimensional lattice polytope. Then we define

X_{P} = X_{(kP)∩M}

where k is any positive integer such that kP is very ample.

Such integers k exist by Corollary 2.17, and if k and ℓ are two such integers, then kP and ℓP have the same normal fan by Proposition 3.9, namely Σ_{kP} = Σ_{ℓP} = Σ_{P}. It follows that while X_{(kP)∩M} and X_{(ℓP)∩M} lie in different projective spaces, they are built from the affine toric varieties U_{σ_{v}} glued together via (3.4). Once we develop the language of abstract varieties in Chapter 3, we will see that X_{P} is well-defined as an abstract variety.

We will often speak of X_{P} without regard to the projective embedding. When we want to use a specific embedding, we will say “X_{P} is embedded using kP ”, where we assume that kP is very ample. In Chapters 5 and 6 we will use the language of divisors and line bundles to restate this in terms of a divisor D_{P} on X_{P} such that kD_{P} is very ample precisely when kP is.

Here is a simple example to illustrate the difference between X_{P} as an abstract variety and X_{P} as sitting is a specific projective space.

Example 3.14. Consider the n-simplex Δ_{n} ⊆ ℝ^{n}. We can define X_{Δ_{n}} using any integer k ≥ 1 since Δ_{n} is normal and hence very ample. The lattice points in kΔ_{n} correspond to the s_{k} = \binom{n+k}{k} monomials of ℂ[t_{1}, …, t_{n}] of total degree ≤ n. This gives an embedding X_{Δ_{n}} ⊆ ℙ^{s_{k}−1}. When k = 1, Δ_{n} ∩ ℤ^{n} = {0, e_{1}, …, e_{n}}, which easily implies that

X_{Δ_{n}} = ℙ^{n}.

The normal fan of Δ_{n} is described in Exercise 3.6. For an arbitrary k ≥ 1, we can regard X_{Δ_{n}} ⊆ ℙ^{s_{k}−1} as the image of the map

ν_{k} : ℙ^{n} ⟶ ℙ^{s_{k}−1}

defined using all monomials of total degree k in ℂ[x_{0}, …, x_{n}] (Exercise 3.6). It follows that this map is an embedding, usually called the Veronese embedding. But when we forget the embedding, the underlying toric variety is just ℙ^{n}.

The Veronese embedding allows us to construct some interesting affine open subsets of ℙ^{n}. Let f ∈ ℂ[x_{0}, …, x_{n}] be nonzero and homogeneous of degree k and write f = ∑_{|α|=k} c_{α}x^{α}. We write the homogeneous coordinates of ℙ^{s−1} as y_{α} for |α| = k. Then L = ∑_{|α|=k} c_{α}y_{α} is a nonzero linear form in the variables y_{α}, so that ℙ^{s_{k}−1} \ V(L) is a copy of ℂ^{s−1} (Exercise 3.6). If follows that

ℙ^{n} \ V(f) ≃ ν_{k}(ℙ^{n}) ∩ (ℙ^{s_{k}−1} \ V(L))

is an affine variety (usually not toric). This shows that ℙ^{n} has a richer supply of affine open subsets than just the open sets U_{i} = ℙ^{n} \ V(x_{i}) considered earlier in the chapter. ⋄

When we explain the Proj construction ℙ^{n} later in the book, we will see the intrinsic reason why ℙ^{n} \ V(f) is an affine open subset of ℙ^{n}.

Example 3.15. The 2-dimensional analog of the rational normal curve C_{n} is the rational normal scroll S_{a,b}, which is the toric variety of the polygon

P_{a,b} = Conv(0, ae_{1}, e_{2}, be_{2} + e_{2}) ⊆ ℝ^{2},

where a, b ∈ ℕ satisfy 1 ≤ a ≤ b. The polygon P = P_{2,4} and its normal fan are pictured in Figure 9.

Figure 9. The polygon of a rational normal scroll and its normal fan

In general, the polygon P_{a,b} has a + b + 2 lattice points and gives the map

(ℂ^{∗})^{2} ⟶ ℙ^{a+b+1}, (s, t) ↦ (1, s, s^{2}, …, s^{a}, t, st, s^{2}t, …, s^{b}t)

such that S_{a,b} = X_{P_{a,b}} is the Zariski closure of the image. To describe the image, we rewrite the map as

ℂ × ℙ^{1} ⟶ ℙ^{a+b+1}, (s, λ, μ) ↦ (λ, sλ, s^{2}λ, …, s^{a}λ, μ, sμ, s^{2}μ, …, s^{b}μ).

When (λ, μ) = (1, 0), the map is s ↦ (1, s, s^{2}, …, s^{a}, 0, …, 0), which is the rational normal curve C_{a} mapped to the first a + 1 coordinates of ℙ^{a+b+1}. In the same way, (λ, μ) = (0, 1) gives the rational normal curve C_{b} mapped to the last b + 1 coordinates of ℙ^{a+b+1}. If we think of these two curves as the “edges” of a scroll, then fixing s gives a point on each edge, and letting (λ, μ) ∈ ℙ^{1} vary gives the line of the scroll connecting the two points. So it really is a scroll!

An important observation is that the normal fan depends only on the difference b − a, since this determines the slope of the slanted edge of P_{a,b}. If we denote the difference by r ∈ ℕ, it follows that as abstract toric varieties, we have

X_{P_{1,r+1}} = X_{P_{2,r+2}} = X_{P_{2,r+2}} = ⋯

since they are all constructed from the same normal fan. In Chapter 3, we will see that this is the Hirzebruch surface H_{r}.

But if we think of the projective surface S_{a,b} ⊆ ℙ^{a+b+1}, then a and b have a unique meaning. For example, they have a strong influence on the defining equations of S_{a,b}. Let the homogeneous coordinates of ℙ^{a+b+1} be x_{0}, …, x_{a}, y_{0}, …, y_{b} and consider the 2 × (a + b) matrix

(\begin{matrix}x_{0} & x_{1} & ⋯ & x_{a−1} \\ x_{1} & x_{1} & ⋯ & x_{a} \end{matrix}| \begin{matrix}y_{0} & y_{1} & ⋯ & y_{b−1} \\ y_{1} & y_{1} & ⋯ & y_{b} \end{matrix}) .

One can show that I(S_{a,b} ⊆ ℂ[x_{0}, …, x_{a}, y_{0}, …, y_{b}] is generated by the 2 × 2 minors of this matrix (see [16, Exercise 9.11], for example). ⋄

Example 3.15 is another example of a determinantal variety, as is the rational normal curve from Example 0.1. Note that the rational normal curve C_{n} comes from the polytope [0, n] = nΔ_{1}, where the underlying toric variety is just ℙ^{1}.

Exercises for § 3.

3.1. This exercise will use the same notation as the proof of Theorem 3.1.

(a) Let H_{u,a} be a supporting hyperplane of a vertex m_{i} ∈ P. Prove that H_{u,0} is a supporting hyperplane of 0 ∈ C_{i}

(b) Prove that dim C_{i} = dim P.

3.2. Consider the maps defined in (3.2).

(a) Show that these maps are inverses of each other and define a bijection between the faces of the cone C_{v} and the faces of P containing v.

(b) Prove that these maps preserve dimensions, inclusions, and intersections.

(c) Explain how this exercise relates to Exercise3.1.

3.3. Prove Proposition 3.9.

3.4. Let P ⊆ M_{ℝ} ≃ ℝ^{n} be an n-dimensional lattice polytope containing 0 as an interior point, and let P^{∘} ⊂ N_{ℝ} be its dual polytope. Prove that the normal fan Σ_{P} consists of the cones over the faces of P^{∘}. Hint: Exercise 2.1 will be useful.

3.5. Let P ⊆ M_{ℝ} be a reflexive polytope.

(a) Prove that 0 is the only interior lattice point of P.

(b) Prove that P^{∘} ⊂ N_{ℝ} is reflexive.

3.6. This exercise is concerned with Example 3.14

(a) Let e_{1}, …, e_{n} be the standard basis of ℝ^{n}. Prove that the normal fan of the standard n-simplex consists of the cones ConeS for all proper subsets S ⊆ {e_{0}, e_{1}, …, e_{n}}. Draw pictures of the normal fan for n = 1, 2, 3.

(b) For an integer k ≥ 1, show that the variety X_{kΔ_{n}} ⊆ ℙ^{s_{k}−1} is given by the map ν_{k} : ℙ^{n} ⟶ ℙ^{s_{k}−1}defined using all monomials of total degree k in ℂ[x_{0}, …, x_{n}].

3.7. Let P ⊆ M_{ℝ} ≃ ℝ^{n} be an n-dimensional lattice polytope and let Q ⊆ P be a face. Prove the following intrinsic description of cone σ_{Q} ∈ Σ_{P}:

σ_{Q} − {u ∈ N_{ℝ} | ⟨m, u⟩ ≤ ⟨m′, u⟩ for all m ∈ Q, m′inP }.

3.8. Prove that all lattice rectangles in the plane with edges parallel to the coordinate axes have the same normal fan.

4. Properties of Projective Toric Varieties

We conclude this section by studying when the projective toric variety X_{P} of a polytope P is smooth or normal.

Normality. For the toric variety X_{P} of a polytope, we have the following basic result on normality.

Theorem 4.1. Let P ⊆ M_{ℝ} ≃ ℝ^{n} be an n-dimensional lattice polytope. Then:

(a) X_{P} is normal.

(b) X_{P} is projectively normal under the embedding given by kP if and only if kP is normal.

Proof. Part (a) is immediate since X_{P} is the union of affine pieces U_{σ_{v}}, v a vertex of P, and U_{σ_{v}} is normal by Theorem 3.5 of Chapter 1. In Chapter 3 we will give an intrinsic definition of normality that will make this argument completely rigorous.

For part (b), the discussion following (1.4)shows that the projective embedding of X_{P} given by X_{(kP)∩M} has affine cone given by Y_{((kP)∩M)×{1}}. By Theorem 3.5 of Chapter 1, this is normal if and only if the semigroup ℕ((kP) ∩ M) × {1} is saturated in M × ℤ, and since ((kP) ∩ M) × {1} generates the cone C_{P}, this is equivalent to saying that (kP)∩M)×{1} generates the semigroup C_{P} ∩(M ×ℤ). Then we are done by Lemma 2.12. □

Smoothness. Given the results of Chapter 1, the smoothness of X_{P} is equally easy to determine. We need one definition.

Definition 4.2. Let P ⊆ M_{ℝ} be a lattice polytope.

(a) Given an a vertex v of P and an edge E containing v, let w_{E} be the first lattice point of E different from v encountered as one tranverses E starting at v. In other words, w_{E} − v is the ray generator of the ray Cone(E − v).

(b) P is smooth if for every vertex v, the vectors w_{E} − v, where E is an edge of P containing v, form a subset of a basis of M. In particular, if dim P = dim M_{ℝ}, then the vectors w_{E} − v form a basis of M.

We can now characterize when X_{P} is smooth.

Theorem 4.3. Let P ⊆ M_{ℝ} ≃ ℝ^{n} be an n-dimensional lattice polytope. Then the following are equivalent:

(a) X_{P} is a smooth projective variety.

(b) Σ_{P} is a smooth fan, meaning that every cone in Σ_{P} is smooth in the sense of Definition 2.14.

(c) P is a smooth polytope.

Proof. Smoothness is a local condition, so that a variety is smooth if and only if its local pieces are smooth. Thus X_{P} is smooth if and only if U_{σ_{v}} is smooth for every vertex v of P, and U_{σ_{v}} is smooth if and only if σ_{v} is smooth. Since faces of smooth cones are smooth and Σ_{P} consists of the σ_{v} and their faces, the equivalence (a) ⇔ (b) follows immediately.

For (b) ⇔ (c), first observe that σ_{v} is smooth if and only if it dual C_{v} = σ_{v}^{∨} is smooth. The discussion following (3.2) makes it easy to see that the ray generators of C_{v} are the vectors w_{E} − v from Definition 4.2. It follows immediately that P is smooth if and only if C_{v} is smooth for every vertex v, and we are done. □

The theorem makes it easy to check the smoothness of simple examples such as the toric variety of the hexagon in Example 3.7 or the rational normal scroll S_{a,b} of Example 3.15 (Exercise 4.1).

We also note the following useful fact, which you will prove in Exercise 4.2

Proposition 4.4. Every smooth n-dimensional lattice polytope P ⊆ M_{ℝ} ≃ ℝ^{n} is very ample. □

Here is an example of a smooth reflexive polytope whose dual is not smooth.

Example 4.5. Let P = (n + 1)Δ_{n} − (1, …, 1) ⊆ ℝ^{n}, where Δ_{n} is the standard n simplex. Proposition 3.9 implies that P and Δ_{n} have the same normal fan, so that P and X_{P} are smooth.

The polytope P has the following further interesting properties (Exercise 4.3). First, P has the facet presentation

x_{i} ≥ −1, i = 1, …, n,
−x_{1} − ⋯ − x_{n} ≥ −1,

so that P is reflexive with dual

P^{∘} = Conv(e_{0}, e_{1}, …, e_{n}), e_{0} = −e_{1} − ⋯ − e_{n}.

The normal fan of P^{∘} consists of cones over the faces of P. In particular, the cone of Σ_{P^{∘}} corresponding to the vertex e_{0} ∈ P^{∘} is the cone

σ_{e_{0}} = Conv(v_{1}, …, v_{n}), v_{i} = e_{0} + (n + 1)e_{i}.

Figure 10 shows P and the cone σ_{e_{0}} when n = 2.

Figure 10. The cone σ_{e_{0}} of the normal fan of P^{∘}

For general n, observe that v_{i} − v_{j} = (n + 1)(e_{i} − e_{j}). This makes it easy to see that ℤv_{1} + ⋯ + ℤv_{n} has index (n + 1)^{n−1} in ℤ^{n}. Thus σ_{e_{0}} is not smooth when n ≥ 2. It follows that the “dual” toric variety X_{P^{∘}} is singular for n ≥ 2. Later we will construct X_{P^{∘}} as the quotient of ℙ^{n} under the action of a finite group G ≃ (ℤ/(n + 1)ℤ)^{n−1}. ⋄

Example 4.6. Consider P = Conv(0, 2e_{1}, e_{2}) ⊆ ℝ^{2}. Since P is very ample, the lattice points P ∩ ℤ^{2} = {0, e_{1}, 2e_{1}, e_{2}} give the map (ℂ^{∗})^{2} → ℙ^{3} defined by

(s, t) ⟼ (1, s, s^{2}, t)

such that X_{P} is the Zariski closure of the image. If ℙ^{3} has homogeneous coordinates y_{0}, y_{1}, y_{2}, y_{3}, then we have

X_{P} = V(y_{0}y_{2} − y_{1}^{2}) ⊆ ℙ^{3}.

Comparing this to Example 0.5, we see that X_{P} is the weighted projective space ℙ(1, 1, 2). Later we will learn the systematic reason why this is true.

The variety X_{P} is not smooth. By working on the affine piece X_{P} ∩ U_{3}, one can check directly that (0, 0, 0, 1) is a singular point of X_{P}.

We can also use Theorem 4.3 and the normal fan of P, shown in Figure 11. One can check that the cones σ_{0} and σ_{1} are smooth, but σ_{2} is not, so that Σ_{P} is

Figure 11. The polygon giving ℙ(1, 1, 2) and its normal fan

not a smooth fan. In terms of P, note that the vectors from v_{2} to the first lattice points along the edges containing v_{2} do not generate ℤ^{2}. Either way, Theorem 4.3 implies that X_{P} is not smooth.

If you look carefully, you will see that σ_{2} is the only nonsmooth cone of the normal fan Σ_{P}. Once we study the correspondence between cones and orbits in Chapter 3, we will see that the non-smooth cone σ_{2} corresponds to the singular point (0, 0, 0, 1) of X_{P}. ⋄

Products of Projective Toric Varieties. Our final task is to understand the toric variety of a product of polytopes. Let P_{i} ⊆ (M_{i})_{ℝ} ≃ ℝ^{n_{i}} be lattice polytopes with dim P_{i} = n_{i} for i = 1, 2. This gives a lattice polytope P_{1} × P_{2} ⊂ (M_{1} × M_{2})_{ℝ} of dimension n_{1} + n_{2}.

Replacing P_{1} and P_{2} with suitable multiples, we can assume that P_{1} and P_{2} are very ample. This gives projective embeddings

X_{P_{i}} ↪ ℙ^{s_{i}−1}, s_{i} = |P_{i} ∩ M_{i}|,

so that by Proposition 0.4, X_{P_{1}} × X_{P_{2}} is a subvariety of ℙ^{s_{1}−1} × ℙ^{s_{1}−1}. Using the Segre embedding

ℙ^{s_{1}−1} × ℙ^{s_{2}−1} ↪ ℙ^{s−1}, s = s_{1}s_{2},

we get an embedding

(4.1)  X_{P_{1}} × X_{P_{2}} ↪ ℙ^{s−1}.

We can understand this projective variety as follows.

Theorem 4.7. If P_{1} and P_{2} are very ample, then

(a) P_{1} × P_{2} ⊂ (M_{1} × M_{2})_{ℝ} is a very ample polytope with lattice points

(P_{1} × P_{2}) ∩ (M_{1} × M_{2}) = (P_{1} ∩ M_{1}) × (P_{2} ∩ M_{2}).

Thus the integer s defined above is s = |(P_{1} × P_{2}) ∩ (M_{1} × M_{2})|.

(b) The image of the embedding X_{P_{1}} × X_{P_{2}} ↪ ℙ^{s−1} coming from the very ample polytope P_{1} × P_{2} equals the image of (4.1).

(c) X_{P_{1}×P_{2}} ≃ X_{P_{1}} × X_{P_{2}}.

Proof. For part (a), the assertions about lattice points are clear. The vertices of P_{1} × P_{2} consist of ordered pairs (v_{1}, v_{2}) where v_{i} is a vertex of P_{i} (Exercise 4.4). Given such a vertex, we have

(P_{1} × P_{2}) ∩ (M_{1} × M_{2}) − (v_{1}, v_{2}) = (P_{1} ∩ M_{1} − v_{1}) × (P_{2} ∩ M_{2} − v_{2}).

Since P_{i} is very ample, we know that ℕ(P_{i} ∩ M_{i} − v_{i}) is saturated in M_{i}. From here, it follows easily that P_{1} × P_{2} is very ample.

For part (b), let T_{N_{i}} be the torus of X_{P_{i}}. Since T_{N_{i}} is Zariski dense in X_{P_{i}}, it follows that T_{N_{1}} × T_{N_{2}} is Zariski dense in X_{P_{1}} × X_{P_{2}} (Exercise 4.4). When combined with the Segre embedding, it follows that X_{P_{1}} × X_{P_{2}} is the Zariski closure of the image of the map

T_{N_{1}} × T_{N_{2}} ⟶ ℙ^{s_{1}s_{2}−1}

given by the characters χ^{m}χ^{m′}, where m ranges over the s_{1} elements of P_{1} ∩ M_{1} and m ranges over the s_{2} elements of P_{2} ∩ M_{2}. When we identify T_{N_{1}} × T_{N_{2}} with T_{N_{1}×N_{2}}, the product χ^{m}χ^{m′} becomes the character χ^{(m,m′)}, so that the above map coincides with the map

T_{N_{×}N_{2}} ⟶ ℙ^{s−1}

coming from the product polytope P_{1} × P_{2} ⊂ (M_{1} × M_{2})_{ℝ}. Part (b) follows, and part (c) is an immediate consequence. □

Here is an obvious example.

Example 4.8. Since ℙ^{n} is the toric variety of the standard n-simplex Δ_{n}, it follows that ℙ^{n} × ℙ^{n} is the toric variety of Δ_{n} × Δ_{m}.

This also works for more than two factors. Thus ℙ^{1} × ℙ^{1} × ℙ^{1} is the toric variety of the cube pictured in Figure 8.

To have a complete theory of products, we need to know what happens to the normal fan. Here is the result, whose proof is left to the reader (Exercise 4.5).

Proposition 4.9. Let P_{i} ⊆ (M_{i})_{ℝ} ≃ ℝ^{n_{i}} be lattice polytopes with dim P_{i} = n_{i} for i = 1, 2. Then

Σ_{P_{1}×P_{2}} = Σ_{P_{1}} ×Σ_{P_{2}}. □

Here is an easy example.

Example 4.10. The normal fan of an interval [a, b] ⊂ ℝ, where a < b in ℤ, is given by

0 σ_{1}  σ_{0}

The corresponding toric variety is ℙ^{1}. The cartesian product of two such intervals is a lattice rectangle whose toric variety is ℙ^{1} × ℙ^{1} by Theorem 4.7. If we set σ_{ij} = σ_{i} × σ_{j}, then Proposition 4.9 gives the normal fan given in Figure 12.

Figure 12. The normal fan of a lattice rectangle giving ℙ^{1} × ℙ^{1}

We will revisit this example in Chapter 3 when we construct toric varieties directly from fans. ⋄

Proposition 4.9 suggests a different way to think about the product. Let v_{i} range over the vertices of P_{i}. Then the σ_{v_{i}} are the maximal cones in the normal fan Σ_{P_{i}}, which implies that

(4.2)  X_{P_{i}} = _{⋃v_{i}}U_{σ_{v_{i}}} .

Thus

X_{P_{1}} × X_{P_{2}} = (_{⋃v_{1}}U_{σ_{v_{1}}} ) × (_{⋃v_{2}}U_{σ_{v_{2}}} )
= _{⋃(v_{1},v_{2})}U_{σ_{v_{1}}} × U_{σ_{v_{2}}}
= _{⋃(v_{1},v_{2})}U_{σ_{v_{1}}×σ_{v_{2}}}
= _{⋃(v_{1},v_{2})}U_{σ_{(v_{1},v_{2})}} = X_{P_{1}×P_{2}}.

In this sequence of equalities, the first follows from (4.2), the second is obvious, the third uses Exercise 3.13 of Chapter 1, the fourth uses Proposition 4.9, and the last follows since (v_{1}, v_{2}) ranges over all vertices of P_{1} × P_{2}.

This argument shows that we can construct cartesian products of varieties using affine open covers, which reduces to the cartesian product of affine varieties defined in Chapter 1. We will use this idea in Chapter 3 to define the cartesian product of abstract varieties.

4. Properties of Projective Toric Varieties 87

Exercises for § 4.

4.1. Show that the hexagon P = Conv(0, e_{1}, e_{2}, 2e_{1} + e_{2}, e_{1} + 2e_{2}, 2e_{1} + 2e_{2}) pictured in Figure 6 and the trapezoid P_{a,b} pictured in Figure 9 are smooth polygons. Also, of the polytopes shown in Figure 8, determine which ones are smooth.

4.2. Prove Proposition 4.4.

4.3. Consider the polytope P = (n + 1)Δ_{n} − (1, …, 1) from Example 4.5.

(a) Verify the facet presentation of P given in the example.

(b) What is the facet presentation of P^{∘}? Hint: You know the vertices of P.

(c) Let v_{i} = e_{0} + (n + 1)e_{i}, where i = 1, …, n and e_{0} = −e_{1} − ⋯ − e_{n}, and let L = ℤv_{1}+⋯+ℤv_{n}. Use the hint given in the text to prove ℤ^{n}/L ≃ (ℤ/(n+1)ℤ)^{n−1}. This shows that the index of L in ℤ^{n} is (n + 1)^{n_{1}}, as claimed in the text.

4.4. Let P_{i} ⊆ (M_{i})_{ℝ} ≃ ℝ^{n_{i}} be lattice polytopes with dim P_{i} = n_{i} for i = 1, 2. Also let S_{i} be the set of vertices of P_{i}.

(a) Use supporting hyperplanes to prove that every element of S_{1} × S_{2} is a vertex of P_{1} × P_{2}.

(b) Prove that P_{1} × P_{2} = Conv(S_{1} × S_{2}) and conclude that S_{1} × S_{2} is the set of vertices of P_{1} × P_{2}.

4.5. The goal of this exercise is to prove Proposition 4.9. We know from Exercise 4.4 that the vertices of P_{1} × P_{2} are the ordered pairs (v_{1}, v_{2}) where v_{i} is a vertex of P_{i}.

(a) Adapt the argument of part (a) of Theorem 4.7 to show that C_{(v_{1},v_{2})} = C_{v_{1}} × C_{v_{2}}. Taking duals, we see that the maximal cones of Σ_{P_{1}×P_{2}} are σ_{(v_{1},v_{2})} = σ_{v_{1}} × σ_{v_{2}}.

(b) Given rational polyhedral cones σ_{i} ⊆ (N_{i})_{ℝ} and faces τ_{i} ⊆ σ_{i}, prove that τ_{1} × τ_{2} is a face of σ_{1} × σ_{2} and that all faces of σ_{1} × σ_{2} arise this way.

(c) Prove that Σ_{P_{1}×P_{2}} = Σ_{P_{1}} × Σ_{P_{2}}.

4.6. Consider positive integers 1 = q_{0} ≤ q_{2} ≤ ⋯ ≤ q_{n} with the property that q_{i}| ∑_{i=0}^{n} q_{i} for i = 0, …, n. Set k_{i} = ( ∑_{i=0}^{n} q_{i})/q_{i} for i = 1, …, n and let

P_{q_{0},...,q_{n}} = Conv(0, k_{1}e_{1}, k_{2}e_{2}, …, k_{n}e_{n}) − (1, …, 1).

Prove that P_{q_{0},...,q_{n}} is reflexive and explain how it relates to Example 4.5. We will prove later that the toric variety of this polytope is the weighted projective space ℙ(q_{0}, …, q_{n}).

4.7. The Sylvester sequence is defined by a_{0} = 2 and a_{k+1} = 1 + a_{1}a_{2} ⋯ a_{k}. It begins 2, 3, 7, 43, 1807, … and is described in [31, A000058]. Now fix a positive integer n ≥ 3 and define q_{0}, …, q_{n} by q_{0} = q_{1} = 1 and q_{i} = 2(a_{n−1} − 1)/a_{n−i} for i = 2, …, n. For n = 3 and 4 this gives 1, 1, 4, 6 and 1, 1, 12, 28, 42. Prove that q_{0}, …, q_{n} satisfies the conditions of Exercise 4.6 and hence gives a reflexive simplex, denoted S_{Q′_{n}} in [25]. This paper proves when n ≥ 4, S_{Q′_{n}} has the largest volume of all n-dimensional reflexive simplices and conjectures that it also has the largest number of lattice points.

Normal Toric Varieties

0. Background: Abstract Varieties

The projective toric varieties studied in Chapter 2 are unions of Zariski open sets, each of which is an affine variety. We will begin this chapter by building up to a general construction of abstract varieties obtained by gluing together affine varieties in an analogous way. The resulting varieties will be abstract in the sense that they do not come with any given ambient affine or projective space. We will see that this is exactly the idea needed to construct a toric variety using the combinatorial data contained in a fan.

Since we make extensive use of localization, we will work exclusively with irreducible affine varieties. Sheaf theory, while important for later chapters, will make only a modest appearance here. For a more general approach to the concept of abstract variety, we recommend standard books such as [12], [17] or [30].

Regular Functions. Let V = Spec(R) be an irreducible affine variety. In §0 of Chapter 1, we defined the Zariski open subset V_{f} = V \ V(f) ⊆ V for f ∈ R and showed that V_{f} = Spec(R_{f}), where R_{f} is the localization of R at f. The open sets V_{f} form a basis for the Zariski topology on V in the sense that every open set U is a (finite) union U = ⋃_{f∈S} V_{f} for some S ⊆ R (Exercise 0.1).

For an affine variety, a morphism V → ℂ is called a regular map, so that the coordinate ring of V consists of all regular maps from V to ℂ. We now define what it means to be regular on an open subset of V.

Definition 0.1. Given an irreducible affine variety V = Spec(R) and a Zariski open U ⊆ V, we say ϕ : U → ℂ is regular if for all p ∈ U, there exists f_{p} ∈ R such that p ∈ V_{f_{p}} ⊆ U and ϕ|_{V_{fp}} ∈ R_{f_{p}}. Then define

O_{V} (U) = {ϕ : U → ℂ | ϕ is regular}.

The condition p ∈ V_{f_{p}} means that f_{p}(p) ≠ 0, and saying ϕ|_{V_{fp}} ∈ R_{f_{p}} means

that ϕ = a_{p}/f_{p}^{n_{p}} for some a_{p} ∈ R and n_{p} ≥ 0.

Here are some cases where O_{V} (U) is easy to compute.

Proposition 0.2. Let V = Spec(R) be an irreducible affine variety.

(a) O_{V} (V) = R.

(b) If f ∈ R, then O_{V} (V_{f}) = R_{f}.

Proof. It is clear from Definition 0.1 that elements of R define regular functions on V, hence elements of O_{V} (V). Conversely, if ϕ ∈ O_{V} (V), then for all p ∈ V there is f_{p} ∈ R such that p ∈ V_{f_{p}} and ϕ = a_{p}/f_{p}^{n_{p}} ∈ R_{f_{p}}. The ideal I = ⟨f_{p}^{n_{p}} | p ∈ V ⟩ ⊆ R satisfies V(I) = ∅ since f_{p}(p) ≠ 0 for all p ∈ V. Hence the Nullstellensatz implies that \sqrt{I} = I(V(I)) = R, so there exists a finite set S ⊆ V and polynomials g_{p} for p ∈ S such that

1 = ∑_{p∈S} g_{p}f_{p}^{n_{p}}.

Hence ϕ = ∑_{p∈S} g_{p}f_{p}^{n_{p}}ϕ = ∑_{p∈S} g_{p} a_{p} ∈ R, as desired.

For part (b), let U ⊆ V_{f} be Zariski open. Then U is Zariski open in V, and whenever g ∈ R satisfies V_{g} ⊆ U, we have V_{g} = V_{fg} with coordinate ring

R_{fg} = (R_{f})_{g/f^{ℓ}}

for all ℓ ≥ 0. These observations easily imply that

(0.1)  O_{V} (U) = O_{V_{f}} (U).

Then setting U = V_{f} gives

O_{V} (V_{f}) = O_{V_{f}} (V_{f}) = R_{f},

where the last equality follows by applying part (a) to V_{f} = Spec(R_{f}). □

Local Rings. When V = Spec(R) is an irreducible affine variety, we can describe regular functions using the local rings O_{V,p} introduced in §0 of Chapter 1. A rational function in ℂ(V) is contained in the local ring O_{V,p} precisely when it is regular in a neighborhood of p. It follows that whenever U ⊆ V is open, we have

⋂_{p∈U} O_{V,p} = O_{V} (U).

Thus regular functions on U are rational functions on V that are defined everywhere on U. In particular, when U = V, Proposition 0.2 implies that

(0.2)  ⋂_{p∈V} O_{V,p} = O_{V} (V) = R = ℂ[V].

The Structure Sheaf. Given an irreducible affine variety V, the operation

U ↦ O_{V} (U), U ⊆ V open,

has the following useful properties:

• When W ⊆ U, Definition 0.1 shows that there is an obvious restriction map

ρ_{U,W} : O_{V} (U) → O_{V} (W)

defined by ρ_{U,W} (ϕ) = ϕ|_{W}. It follows that ρ_{U,U} is the identity map and that ρ_{V,W} ∘ ρ_{U,V} = ρ_{U,W} whenever W ⊆ V ⊆ U.

• If {U_{α}} is an open cover of U ⊆ V, then the sequence

0 ⟶ O_{V} (U) ⟶ ∏_{α} O_{V} (U_{α}) \overrightarrow{⟶} ∏_{α,β} O_{V} (U_{α} ∩ U_{β})

is exact. Here, the second arrow is defined by the restrictions ρ_{U,U_{α}} and the double arrow is defined by ρ_{U_{α},U_{α}∩U_{β}} and ρ_{U_{β},U_{α}∩U_{β}}. Exactness at O_{V} (U) means that regular functions are determined locally (that is, two regular functions on U are equal if their restrictions to all U_{α} are equal), and exactness at ∏_{α} O_{V} (U_{α}) means that regular functions on the U_{α} agreeing on the overlaps U_{α} ∩ U_{β} patch together to give a regular function on U.

In the language of sheaf theory, these properties imply that O_{V} is a sheaf of ℂalgebras, called the structure sheaf of V. We call (V, O_{V}) a ringed space over ℂ. Also, since (0.1) holds for all open sets U ⊆ V_{f}, we write

O_{V} |_{V_{f}} = O_{V_{f}} .

In terms of ringed spaces, this means (V_{f}, O_{V} |_{V_{f}}) = (V_{f}, O_{V_{f}}).

Morphisms. By §0 of Chapter 1, a polynomial mapping F : V_{1} → V_{2} between affine varieties corresponds to the ℂ-algebra homomorphism F^{∗} : ℂ[V_{2}] → ℂ[V_{1}] defined by F^{∗}(g) = g ∘ F for g ∈ ℂ[V_{2}]. We now extend this to open sets of affine varieties.

Definition 0.3. Let U_{i} ⊆ V_{i} be Zariski open subsets of irreducible affine varieties for i = 1, 2. A function Φ : U_{1} → U_{2} is a morphism if ϕ ↦ ϕ ∘ Φ defines a map

Φ^{∗} : O_{V_{2}}(U_{2}) ⟶ O_{V_{1}}(U_{1}).

Thus Φ : U_{1} → U_{2} is a morphism if composing Φ with regular functions on U_{2} gives regular functions on U_{1}. Note also that Φ^{∗} is a ℂ-algebra homomorphism since it comes from composition of functions.

Example 0.4. Suppose that Φ : V_{1} → V_{2} is a morphism according to Definition 0.3. If V_{i} = Spec(R_{i}), then the above map Φ^{∗} gives the ℂ-algebra homomorphism

R_{2} = O_{V_{2}}(V_{2}) ⟶ O_{V_{1}}(V_{1}) = R_{1}.

By Chapter 1, the homomorphism R_{2} → R_{1} gives a map of affine varieties V_{2} → V_{1}. In Exercise 0.3 you will show that this is the map Φ : V_{1} → V_{2} we started with.

⋄

Example 0.4 shows that when we apply Definition 0.3 to maps between affine varieties, we get the same morphisms as in Chapter 1. In Exercise 0.3 you will verify the following properties of morphisms:

• If U is open in an affine variety V, then

O_{V} (U) = {ϕ : U → ℂ | ϕ is a morphism}.

Hence regular functions on U are just morphisms from U to ℂ.

• A composition of morphisms is morphism.

• An inclusion of open sets W ⊆ U of an affine variety V is a morphism. These observations enable us to understand what it means for Φ : U_{1} → U_{2} be

a morphism when U_{i} ⊆ V_{i} are arbitrary open sets. Let V_{i} = Spec(R_{i}) and write U_{1} = ⋃_{f∈S}(V_{1})_{f} for S ⊆ R_{1} finite. Then (V_{1})_{f} ⊆ U_{1} and U_{2} ⊆ V_{2} gives the morphism

(V_{1})_{f} ⊆ U_{1} ⟶^{Φ} U_{2} ⊆ V_{2}.

This is given by a ℂ-algebra homomorphism R_{2} → (R_{1})_{f}. Since the (V_{1})_{f} cover U_{1}, we see that locally, morphisms look like the polynomial maps introduced in Chapter 1. In particular, morphisms are continuous in the Zariski topology.

We say that a morphism Φ : U_{1} → U_{2} is an isomorphism if Φ is bijective and its inverse function Φ^{−1} : U_{2} → U_{1} is also a morphism.

Gluing Together Affine Varieties. We now are ready to define abstract varieties by gluing together open subsets of affine varieties. The model is what happens for ℙ^{n}. Recall from §0 of Chapter 2 that ℙ^{n} is covered by open sets

U_{i} = ℙ^{n} \ V(x_{i}) = Spec(ℂ[\frac{x_{1}}{x_{i}} , …, \frac{x_{i−1}}{x_{i}} , \frac{x_{i+1}}{x_{i}} , …, \frac{x_{n}}{x_{i}} ])

for i = 0, …, n. Each U_{i} is a copy of ℂ^{n} that uses a different set of variables. Given i ≠ j, these copies of ℂ^{n} are “glued together” as follows. We have open subsets

(0.3)  (U_{i})_{\frac{x_{j}}{x_{i}}} ⊆ U_{i} and (U_{j})_{\frac{x_{i}}{x_{j}}} ⊆ U_{j},

and we also have the isomorphism

(0.4)  g_{ij} : (U_{i})_{\frac{x_{j}}{x_{i}}} ≃ (U_{j})_{\frac{x_{i}}{x_{j}}}

since both give the same open set U_{i} ∩ U_{j} in ℙ^{n}. At the level of coordinate rings, this comes from the isomorphism

g_{ij}^{∗} : ℂ[\frac{x_{1}}{x_{j}} , …, \frac{x_{j−1}}{x_{j}} , \frac{x_{j+1}}{x_{j}} , …, \frac{x_{n}}{x_{j}} ]_{\frac{x_{i}}{x_{j}}} ≃ ℂ[\frac{x_{1}}{x_{i}} , …, \frac{x_{i−1}}{x_{i}} , \frac{x_{i+1}}{x_{i}} , …, \frac{x_{n}}{x_{i}} ]_{\frac{x_{j}}{x_{i}}} defined by

\frac{x_{k}}{x_{j}} ⟼ \frac{x_{k}}{x_{i}} /\frac{x_{j}}{x_{i}} (k ≠ j) and (\frac{x_{i}}{x_{j}} )^{−1} ⟼ \frac{x_{j}}{x_{i}} .

We can turn this around and start from the affine varieties U_{i} ≃ ℂ^{n} given above and glue together the open sets in (0.3)using the isomorphisms g_{ij} from (0.4). This gluing is consistent since g_{ji} = g_{ij}^{−1} and g_{ik} = g_{ij} ∘ g_{jk} wherever all three maps are defined. The result of this gluing is the projective space ℙ^{n}.

To generalize this, suppose we have a finite collection {V_{α}}_{α} of irreducible affine varieties and for all pairs α, β we have Zariski open sets V_{αβ} ⊆ V_{α} and isomorphisms g_{αβ} : V_{αβ} ≃ V_{βα} satisfying the following compatibility conditions:

• g_{βα} = g_{αβ}^{−1} for all pairs α, β.

• g_{αβ}(V_{αβ} ∩ V_{αγ}) = V_{βα} ∩ V_{βγ} and g_{αγ} = g_{βγ} ∘ g_{αβ} on V_{αβ} ∩ V_{αγ} for all triples α, β, γ. We are now ready to glue. Let Y be the disjoint union of the V_{α} and define a

relation ∼ on Y by a ∼ b if and only if a ∈ V_{α}, b ∈ V_{β} for some α, β with b = g_{αβ}(a). The first compatibility condition shows that ∼ is reflexive and symmetric; the second shows that it is transitive. Hence ∼ is an equivalence relation and we can form the quotient space X = Y/ ∼ with the quotient topology. For each α, let

U_{α} = {[a] ∈ X | a ∈ V_{α}}.

Then U_{α} ⊆ X is an open set and the map h_{α}(a) = [a] defines a homeomorphism h_{α} : V_{α} ≃ U_{α} ⊆ X. Thus X locally looks like an irreducible affine variety.

Definition 0.5. We call X the abstract variety determined by the above data.

An abstract variety X comes equipped with the Zariski topology whose open sets are those sets that restrict to open sets in each U_{α}. The Zariski closed subsets Y ⊆ X are called subvarieties of X. We say that X is irreducible is it is not the union of two proper subvarieties. Since X is locally irreducible, it is irreducible if and only if it is connected.

The Structure Sheaf of an Abstract Variety. Let U be an an open subset of an abstract variety X and set W_{α} = h_{α}^{−1}(U ∩ U_{a}) ⊆ V_{α}. Then a function ϕ : U → ℂ is regular if

ϕ ∘ h_{α}|_{W_{α}} : W_{α} ⟶ ℂ

is regular for all α. The compatibility conditions ensure that this is well-defined, so that one can define

O_{X}(U) = {ϕ : U → ℂ | ϕ is regular}.

This gives the structure sheaf O_{X} of X. Thus an abstract variety is really a ringed space (X, O_{X}) with a finite open covering {U_{α}}_{α} such that (U_{α}, O_{X}|_{U_{α}}) is isomorphic to the ringed space (V_{α}, O_{V_{α}}) of the affine variety V_{α}. (We leave the definition of isomorphism of ringed spaces to the reader.)

Example 0.6. We saw above that ℙ^{n} can be obtained by gluing together the open sets (0.3) using the isomorphisms g_{ij} from (0.4). This shows that ℙ^{n} is an abstract variety with affine open subset U_{i} ⊆ ℙ^{n}. More generally, given an irreducible projective variety V ⊆ ℙ^{n}, we can cover V with affine open subset V ∩ U_{i}, and the gluing implicit in equation (0.7) from Chapter 2. We conclude that irreducible projective varieties are also abstract varieties.

In a similar way, ℙ^{n} × ℂ^{m} can be viewed as gluing affine spaces U_{i} × ℂ^{m} ≃ ℂ^{n+m} along suitable open subsets. Thus ℙ^{n} × ℂ^{m} is an abstract variety, and the same is true for irreducible subvarieties V ⊆ ℙ^{n} × ℂ^{m}. ⋄

In fact, all of the various types of “irreducible variety” introduced so far in this book can be subsumed under the concept of “irreducible abstract variety.” Even more generally, one can define abstract varieties that locally look like arbitrary affine varieties, not necessarily irreducible (see [12, 17, 30]).

Example 0.7. Consider V_{0} = ℂ^{2} = Spec(ℂ[u, v]) and V_{1} = ℂ^{2} = Spec(ℂ[w, z]) with

V_{01} = V_{0} \ V(v) = Spec(ℂ[u, v]_{v})
V_{10} = V_{1} \ V(z) = Spec(ℂ[w, z]_{z})

and gluing data

g_{01} : V_{01} → V_{10} coming from the ℂ-algebra homomorphism
g_{01}^{∗} : ℂ[w, z]_{z} → ℂ[u, v]_{v} defined by w ↦ uv and z ↦ 1/v

and

g_{10} : V_{10} → V_{01} coming from the ℂ-algebra homomorphism
g_{10}^{∗} : ℂ[u, v]_{v} → ℂ[w, z]_{z} defined by u ↦ wz and v ↦ 1/z.

One checks that g_{01} = g_{10}^{−1}, and the other compatibility condition is satisfied since there are only two V_{i}. It follows that we get an abstract variety X.

The variety X has another description. Consider the product ℙ^{1} × ℂ^{2} with homogeneous coordinates (x_{0}, x_{1}) on ℙ^{1} and coordinates (x, y) on ℂ^{2}. We will identify X with the subvariety W = V(x_{0}y −x_{1}x) ⊆ ℙ^{1} ×ℂ^{2}, called the blow-up of ℂ^{2} at the origin, and denoted Bl_{0}(ℂ^{2}). First note that ℙ^{1} × ℂ^{2} is covered by

U_{0} × ℂ^{2} = Spec(ℂ[x_{1}/x_{0}, x, y]) and U_{1} × ℂ^{2} = Spec(ℂ[x_{0}/x_{1}, x, y]).

Then W is covered by W_{0} = W ∩ (U_{0} × ℂ^{2}) and W_{1} = W ∩ (U_{1} × ℂ^{2}). Also,

W_{0} = V(y − (x_{1}/x_{0})x) ⊆ U_{0} × ℂ^{2},

which gives the coordinate ring

ℂ[x_{1}/x_{0}, x, y]/⟨y − (x_{1}/x_{0})x⟩ ≃ ℂ[x, x_{1}/x_{0}] via y ↦ (x_{1}/x_{0})x.

Similarly, W_{1} = V(x − (x_{0}/x_{1})y) ⊆ U_{1} × ℂ^{2} has coordinate ring

ℂ[x_{0}/x_{1}, x, y]/⟨x − (x_{0}/x_{1})y⟩ ≃ ℂ[y, x_{0}/x_{1}] via x ↦ (x_{0}/x_{1})y.

You should check that these are glued together in W in exactly the same way that V_{0} and V_{1} are glued together in X. We will generalize this example in Exercise 0.8.

⋄

From now on, when considering varieties, we will usually be thinking of abstract varieties. Hence we will usually say “variety” rather than “abstract variety.”

The Local Rings of a Variety. The definition of local ring of an irreducible affine variety extends easily to the abstract case. A rational function on an irreducible variety X is a regular function f : U → ℂ defined on a nonempty Zariski open set of X, and two rational functions on X are equivalent if they agree on a nonempty Zariski open subset. In Exercise 0.4 you will show that this relation is an equivalence relation and that the set of equivalence classes is a field, called the function field of X, denoted ℂ(X).

A rational function f ∈ ℂ(X) is defined at p ∈ X if f is equivalent to a regular function g : U → ℂ such that p ∈ U.

Definition 0.8. Let p be a point of an irreducible variety X. Then

O_{X,p} = {ϕ ∈ ℂ(X) | ϕ is defined atp}.

is the local ring of X at p.

It is not difficult to see that O_{X,p} is a local ring with unique maximal ideal

𝔪_{X,p} = {ϕ ∈ O_{X,p} | ϕ(p) = 0}.

The local ring O_{X,p} can also be defined for general (possibly reducible) varieties as the direct limit

O_{X,p} = lim_{\overrightarrow{p∈U}} O_{X}(U),

over all neighborhoods of p in X. See [12] or [17].

Smooth Varieties and Orbifolds. Because the definition of smoothness at a point p for an affine variety V depends only on the local ring O_{V,p} (see Definition 0.7 from Chapter 1), we can extend this notion to general varieties using the local rings as defined above (see Exercise 0.2).

Definition 0.9. Let X be a variety. We say X is smooth if every point p ∈ X has an affine open neighborhood that is a smooth affine variety.

A similar comment lets us extend the notion of finite quotient singularities to a general variety and to define orbifolds in this more general context.

Morphisms of Varieties. Given varieties X and Y with affine open covers X = ⋃_{α} U_{α} and Y = ⋃_{β} U′_{β}, a function Φ : X → Y is a morphism if it is continuous in the Zariski topology and the restrictions

Φ|_{U_{α}∩Φ^{−1}(U′_{β})} : U_{α} ∩ Φ^{−1}(U′_{β}) ⟶ U′_{β}

are morphisms in the sense of Definition 0.3. One can check that this definition is indpendent of the affine open covers. Also, a function ϕ : X → ℂ is a morphism if and only if it gives an element of O_{X}(X), i.e., ϕ is a regular function.

Normal Varieties. We return to the notion of normality introduced in Chapter 1.

Definition 0.10. An irreducible variety X is called normal if the local rings O_{X,p} are normal for all p ∈ X.

At first glance, this appears to be different from the definition given for affine varieties in Definition 0.3 of Chapter 1. In fact the two notion are equivalent in the affine case.

Proposition 0.11. Let V be an irreducible affine variety. Then ℂ[V] is normal if and only if the local rings O_{V,p} are normal for all p ∈ V.

Proof. If O_{V,p} is normal for all p, then (0.2) shows that ℂ[V] is an intersection of normal domains, all of which have the same field of fractions. Since such an intersection is normal by Exercise 0.5 of Chapter 1, it follows that ℂ[V] is normal.

For the converse, suppose that ℂ[V] is normal and let α ∈ ℂ(V) satisfy

α^{k} + a_{1}α^{k−1} + ⋯ + a_{k} = 0, a_{i} ∈ O_{V,p}.

Write a_{i} = g_{i}/f_{i} with g_{i}, f_{i} ∈ ℂ[V] and f_{i}(p) ≠ 0. The product f = f_{1} ⋯ f_{k} has the properties that a_{i} ∈ ℂ[V]_{f} and f(p) ≠ 0. The localization ℂ[V]_{f} is normal by Exercise 0.5 of Chapter 1 and is contained in O_{V,p} since f(p) ≠ 0. Hence α ∈ ℂ[V]_{f} ⊆ O_{V,p}. This completes the proof. □

We have the following consequence of Proposition 0.11 and Definition 0.10.

Proposition 0.12. Let X be an irreducible variety with a cover consisting of affine open sets V_{α}. Then X is normal if and only if each V_{α} is normal.

Products of Varieties. As another example of abstract varieties and gluing, we indicate why the product X_{1} × X_{2} of varieties X_{1} and X_{2} also has the structure of a variety. In §0 of Chapter 1 we constructed the product of affine varieties. From here, it is relatively routine to see that if X_{1} is obtained by gluing together affine varieties U_{α} and X_{2} is obtained by gluing together affines U′_{β}, then X_{1} × X_{2} is obtained by gluing together the U_{α}×U′_{β} in the corresponding fashion. Furthermore, X_{1} × X_{2} has the correct universal mapping property. Namely, given a diagram

[figure: W; ϕ_{1}; ϕ_{2}; ν; X_{1} × X_{2}; π_{1}; π_{2}; X_{1}; X_{2}]

where ϕ_{i} : W → X_{i} are morphisms, there is a unique morphism ν : W → X_{1}×X_{2} (the dotted arrow) that makes the diagram commute.

The details of products and their properties are presented in [12] and [17].

Example 0.13. Let us construct the product ℙ^{1} × ℂ^{2}. Write ℙ^{1} = V_{0} ∪ V_{1} where V_{0} = Spec(ℂ[u]) and V_{1} = Spec(ℂ[v]), with the gluing given by

ℂ[v]_{v} ≃ ℂ[u]_{u}, v ↦ 1/u.

Then ℙ^{1} × ℂ^{2} is constructed from

U_{0} × ℂ^{2} = Spec(ℂ[u] ⊗ ℂ[x, y]) ≃ ℂ^{3}
U_{1} × ℂ^{2} = Spec(ℂ[v] ⊗ ℂ[x, y]) ≃ ℂ^{3},

with gluing given by

(U_{0} × ℂ^{2})_{u} ≃ (U_{1} × ℂ^{2})_{v}

corresponding to the obvious isomorphism of coordinate rings. ⋄

Separated Varieties. From the point of view of the classical topology, arbitrary gluings can lead to varieties with some strange properties.

Example 0.14. In Example 0.13 we saw how to construct ℙ^{1} from affine varieties V_{0} = Spec(ℂ[u]) ≃ ℂ and V_{1} = Spec(ℂ[v]) ≃ ℂ with the gluing given by v ↦ 1/u on open sets ℂ^{∗} ≃ (V_{0})_{u} ⊆ V_{0} and ℂ^{∗} ≃ (V_{1})_{v} ⊆ V_{1}. This expresses ℙ^{1} as consisting of ℂ^{∗} plus two additional points. But now consider the abstract variety arising from the gluing map

(V_{0})_{u} ⟶ (V_{1})_{v}

that corresponds to the map of ℂ-algebras defined by v ↦ u. As before, the glued variety X consists of ℂ^{∗} together with two additional points. However here we have a morphism π : X → ℂ whose fiber π^{−1}(a) over a ∈ ℂ^{∗} contains one point, but whose fiber over 0 consists of two points, p_{1} corresponding to 0 ∈ V_{0} and p_{2} corresponding to 0 ∈ V_{1}. If U_{1}, U_{2} are classical open sets in X with p_{1} ∈ U_{1} and p_{2} ∈ U_{2}, then U_{1} ∩ U_{2} ≠ ∅. So the classical topology on X is not Hausdorff. ⋄

Since varieties are rarely Hausdorff in the Zariski topology (Exercise 0.5), we need a different way to understand Example 0.14. Consider the product X ×X and the diagonal mapping Δ : X → X × X defined by Δ(p) = (p, p) for p ∈ X. For X from Example 0.14, there is a morphism X × X → ℂ whose fiber over over 0 consists of the four points (p_{i}, p_{j}). Any Zariski closed subset of X × X containing one of these four points must contain all of them. The image of the diagonal mapping contains (p_{1}, p_{1}) and (p_{2}, p_{2}), but not the other two, so the diagonal is not Zariski closed. This example motivates the following definition.

Definition 0.15. We say a variety X is separated if the image of the diagonal map Δ : X → X × X is Zariski closed in X × X.

For instance, ℂ^{n} is separated because the image of the diagonal in ℂ^{n} × ℂ^{n} = Spec(ℂ[x_{1}, …, x_{n}, y_{1}, …, y_{n}]) is the affine variety V(x_{1} − y_{1}, …, x_{n} − y_{n}). Similarly any affine variety is separated.

The connection between failure of separatedness and failure of the Hausdorff property in the classical topology seen in Example 0.14 is a general phenomenon.

Theorem 0.16. A variety is separated if and only if it is Hausdorff in the classical topology. □

Here are some additional properties of separated varieties (Exercise 0.6).

Proposition 0.17. Let X be a separated variety.

(a) If f, g : Y → X are morphisms, then {y ∈ Y | f(y) = g(y)} is a subvariety of Y.

(b) If U, V are affine open subsets of X, then U ∩ V is also affine. □

The requirement that X be separated is often included in the definition of an abstract variety. When this is done, what we have called a variety is sometimes called a pre-variety.

Fibered Products. Finally in this section, we will discuss fibered products of varieties, a construction required for the discussion of proper morphisms in §4. First, if we have mappings of sets f : X → S and g : Y → S, then the fibered product X ×_{S} Y is defined to be

X ×_{S} Y = {(x, y) ∈ X × Y | f(x) = g(y)}.

The fibered product construction gives a very flexible language for describing ordinary products, intersections of subsets, fibers of mappings, the set where two mappings agree and so forth:

• If S is a point, then X ×_{S} Y is the ordinary product X × Y.

• If X and Y are subsets of S and f, g are the inclusion mappings then X×_{S}Y = X ∩ Y.

• If Y = {s} ⊆ S, then X ×_{S} Y = f^{−1}({s}).

• If X = Y, then X ×_{S} X = {(x, x′) | f(x) = g(x′)}, (compare with Proposition 0.17).

This third property is the reason for the name. All are easy exercises that we leave for the reader.

In analogy with the universal mapping property of the product discussed above, we have the following statement. Whenever we have mappings ϕ_{1} : W → X and ϕ_{2} : W → Y such that f ∘ ϕ_{1} = g ∘ ϕ_{2}, there is a unique ν : W → X ×_{S} Y that makes the following diagram commute.

[figure: W; ϕ_{1}; ϕ_{2}; ν; X ×_{S} Y; π_{1}; π_{2}; X; f; Y; g; S]

If X = Spec(R_{1}), Y = Spec(R_{2}), and S = Spec(R) are affine varieties, then the fibered product also exists as an affine variety. Namely, the morphisms f, g correspond to ring homomorphisms

f^{∗} : R → R_{1}
g^{∗} : R → R_{2}

Hence both R_{1}, R_{2} have the structure of R-modules, and we have the tensor product R_{1} ⊗_{R} R_{2}. This is also a finitely generated ℂ-algebra without nilpotents, so it is the coordinate ring of an affine variety. So we take

X ×_{S} Y = Spec(R_{1} ⊗_{R} R_{2})

as the definition of X ×_{S} Y. When S, X, Y are general varieties, X ×_{S} Y is constructed by the gluing procedure. We will not present the details here; interested readers should consult the discussion of the general construction within the theory of schemes given in [12, 17].

Exercises for § 0.

0.1. Let V = Spec(R) be an affine variety.

(a) Show that every ideal I ⊆ R can be written in the form I = ⟨f_{1}, …, f_{s}⟩, where f_{i} ∈ R. (This is the Hilbert Basis Theorem in R.)

(b) Let W ⊆ V be a subvariety. Show that the complement of W in V can be written as a union of a finite collection of open affine sets of the form V_{f}.

(c) Deduce that every open cover of V (in the Zariski topology) has a finite subcover. (This says that affine varieties are quasicompact in the Zariski topology.)

0.2. As in the affine case, we want to say a variety X is smooth at p if dim T_{p}(X) = dim X. In this exercise, you will show that this is a well-defined notion.

100 3. Normal Toric Varieties

(a) Show that if p ∈ X is in the intersection of two affine open sets V_{α} ∩ V_{β}, then the Zariski tangent spaces T_{V_{α},p} and T_{V_{β},p} are isomorphic as vector spaces over ℂ.

(b) Show that dim X is a well-defined integer.

(c) Deduce that the proposed notion of smoothness at p is well-defined.

0.3. This exercises explores some properties of the morphisms described in Definition 0.3.

(a) Prove the claim made in Example 0.4. Hint: Take p ∈ V_{1} and set 𝔪_{p} = {f ∈ R_{1} | f(p) = 0}. Then describe (Φ^{∗})^{−1}(𝔪_{p}) in terms of Φ(p).

(b) Prove the properties of morphisms listed on page 92.

0.4. Let X be an irreducible abstract variety.

(a) Let f, g be rational functions on X. Show that f ∼ g if f|_{U} = g|_{U} for some nonempty open set U ⊆ X is an equivalence relation.

(b) Show that the set of equivalence classes of the relation in part (a) is a field.

(c) Show that if U ⊆ X is an affine open subset of X, then ℂ(U) ≃ ℂ(X).

0.5. Show that a variety is Hausdorff in the Zariski topology if and only if it is a finite set.

0.6. Consider Proposition 0.17.

(a) Prove part (a) of the proposition. Hint: Show first that if F : Y → X × X is defined by F (y) = (f(y), g(y)), then Z = F^{−1}(Δ(X)).

(b) Prove part (b) of proposition. Hint: Show first that U ∩ V is identified with Δ(X) ∩ (U × V) ⊆ X × X.

0.7. Let V = Spec(R) be an affine variety. Which ℂ-algebra homomorphism R ⊗_{ℂ} R → R corresponds to the diagonal mapping Δ : V → V × V? Prove your assertion. Hint: Consider the universal mapping property of V × V.

0.8. In this exercise, we will study an important variety in ℙ^{n−1} × ℂ^{n}, the blow-up of ℂ^{n} at the origin, denoted Bl_{0}(ℂ^{n}). This generalizes Bl_{0}(ℂ^{2}) from Example 0.7. Write the homogeneous coordinates on ℙ^{n−1} as (x_{0}, …, x_{n−1}), and the affine coordinates on ℂ^{n} as (y_{1}, …, y_{n}). Let

(0.5)  W = Bl_{0}(ℂ^{n}) = V(x_{i−1}y_{j} − x_{j−1}y_{i} | 1 ≤ i < j ≤ n) ⊆ ℙ^{n−1} × ℂ^{n}.

Let U_{i−1}, i = 1, …, n be the standard affine opens in ℙ^{n−1}:

U_{i−1} = ℙ^{n−1} \ V(x_{i−1}),

i = 1, …, n (note the slightly non-standard indexing). So the U_{i−1} × ℂ^{n} form a cover of ℙ^{n−1} × ℂ^{n}.

(a) Show that for each i = 1, …, n, W_{i−1} = W ∩ (U_{i−1} × ℂ^{n}) ≃

Spec (ℂ [\frac{x_{0}}{x_{i−1}} , …, \frac{x_{i−2}}{x_{i−1}} , \frac{x_{i}}{x_{i−1}} , …, \frac{x_{n−1}}{x_{i−1}} , y_{i}])

using the equations (0.5) defining W.

(b) Give the gluing data for identifying the subsets W_{i−1} \V(x_{j−1}) and W_{j−1} \V(x_{i−1}).

1. Fans and Normal Toric Varieties

In this section we present the construction of the toric variety X_{Σ} corresponding to a fan Σ. We will also relate the varieties obtained to many examples we have encountered previously, and we will see how properties of the fan correspond to properties such as smoothness and compactness of X_{Σ}.

The Toric Variety Associated to a Fan. A toric variety in this context means the same thing as in Chapters 1 and 2, although we now allow abstract varieties as in §0.

Definition 1.1. A toric variety is an irreducible variety X containing a torus T_{N} ≃ (ℂ^{∗})^{n} as a Zariski open subset such that the action of T_{N} on itself extends to an action of T_{N} on X.

The other ingredient in this section is a fan in the vector space N_{ℝ}.

Definition 1.2. A fan Σ in N_{ℝ} is a finite collection of cones σ such that

(a) every σ in Σ is a strongly convex rational polyhedral cone,

(b) for all σ in Σ, each face of σ is also in Σ, and

(c) if σ_{1} and σ_{2} are in Σ, then the intersection σ_{1} ∩ σ_{2} is a face of each (hence also in Σ).

It is also possible to consider infinite collections of cones with the same properties, but we will not do so in this book.

Examples of fans include the normal fans of lattice polytopes P ⊆ M_{ℝ} as in Chapter 2, §3, and the properties given in this definition were verified for those fans in Corollary 3.5 in that section.

However, there exist fans that are not equal to the normal fan of any lattice polytope. The construction of such fans will be easier to motivate after we have discussed the connections between polytopes, divisors, and rational mappings from toric varieties to ℙ^{n}. We will postpone discussing them until Chapter 6.

We now show how the cones in any fan give the combinatorial data necessary to glue a collection of affine toric varieties together to yield an abstract toric variety. By Chapter 1, Theorem 2.16, for each cone σ in Σ, we have the affine toric variety

U_{σ} = Spec(ℂ[S_{σ}]) = Spec(ℂ[σ^{∨} ∩ M]).

Moreover, if τ is a face of σ, then by Chapter 1, Definition 2.5, there is some m ∈ σ^{∨} such that τ = σ ∩ H_{m}, where H_{m} = {u ∈ N_{ℝ} | ⟨m, u⟩ = 0} is the hyperplane defined by m. In Proposition 3.16 of Chapter 1, we proved the following facts:

• If m ∈ M and τ = σ ∩ H_{m} is the corresponding face of σ, then

(1.1)  S_{τ} = S_{σ} + ℤ(−m),

and ℂ[S_{τ}] is the localization ℂ[S_{σ}]_{χ^{m}}. Therefore, U_{τ} = (U_{σ})_{χ^{m}}.

• If τ = σ_{1} ∩ σ_{2}, then there exists m ∈ σ_{1}^{∨} ∩ (−σ_{2})^{∨} ∩ M such that

σ_{1} ∩ H_{m} = τ = σ_{2} ∩ H_{m}

and

(1.2)  U_{σ_{1}} ⊇ (U_{σ_{1}})_{χ^{m}} = U_{τ} = (U_{σ_{2}})_{χ^{−m}} ⊆ U_{σ_{2}}.

These facts were also used to study projective toric varieties in Chapter 2. The following proposition gives an additional property of the S_{σ} and their semigroup rings that we will need in this section.

Proposition 1.3. If σ_{1}, σ_{2} are rational polyhedral cones such that τ = σ_{1} ∩ σ_{2} is a face of each σ_{i}, then

S_{τ} = S_{σ_{1}} + S_{σ_{2}}.

Proof. The inclusion S_{σ_{1}} + S_{σ_{2}} ⊆ S_{τ} follows directly from the general fact that σ_{1}^{∨} + σ_{2}^{∨} = (σ_{1} ∩ σ_{2})^{∨} = τ^{∨}. The reverse inclusion then follows from the bulleted points above. If p ∈ S_{τ}, then by (1.1), p = q + ℓ(−m) for some q ∈ S_{σ_{1}} and ℓ ∈ ℤ_{≥0}. Since −m ∈ (−σ_{2})^{∨} implies m ∈ S_{σ_{2}}, we have p ∈ S_{σ_{1}} + S_{σ_{2}}. □

This result is sometimes called the separation lemma, and it will be the key ingredient in showing that the toric varieties X_{Σ} are separated using Definition 0.15.

Example 1.4. Let σ_{1} = Cone(e_{1} + e_{2}, e_{2}) (as in Exercise 2.8 of Chapter 1), and let σ_{2} = Cone(e_{1}, e_{1} + e_{2}) in N_{ℝ} ≃ ℝ^{2}. Then τ = σ_{1} ∩ σ_{2} = Cone(e_{1} + e_{2}).

Figure 1. The cones σ_{1}, σ_{2}, τ and their duals

We show the dual cones σ_{1}^{∨} = Cone(e_{1}, −e_{1} + e_{2}), σ_{2}^{∨} = Cone(e_{1} − e_{2}, e_{2}), and τ^{∨} = σ_{1}^{∨} + σ_{2}^{∨} in Figure 1.

The darkest shaded region on the right is σ_{1}^{∨} ∩ σ_{2}^{∨}. We have τ = σ_{1} ∩ H_{m} = σ_{2} ∩ H_{−m}, where m = −e_{1} + e_{2} ∈ σ_{1}^{∨} and −m = e_{1} − e_{2} ∈ σ_{2}^{∨}. Note that S_{τ} equals the set of all sums m + m′ with m ∈ σ_{1}^{∨} ∩ M and m′ ∈ σ_{2}^{∨} ∩ M. Hence S_{τ} = S_{σ_{1}} + S_{σ_{2}}. ⋄

Now consider the collection of affine toric varieties U_{σ} = Spec(ℂ[S_{σ}]), where σ runs over all cones in a fan Σ. Let σ_{1} and σ_{2} be any two of these cones and let τ = σ_{1} ∩ σ_{2}. By (1.2), we have an isomorphisms

g_{σ_{1},σ_{2}} : (U_{σ_{1}})_{χ^{m}} ≃ (U_{σ_{2}})_{χ^{−m}}.

(In fact this mapping is just the identity on U_{τ}.) In Exercise 1.1, you will show that the compatibility conditions as in §0 for gluing the affine varieties U_{σ} along the subvarieties (U_{σ})_{χ^{m}} are satisfied. Hence, we obtain an abstract variety X_{Σ} associated to the fan Σ.

Theorem 1.5. Let Σ be a fan in N_{ℝ}. The variety X_{Σ} is a normal, separated toric variety.

Proof. First, X_{Σ} is irreducible because all of the U_{σ} are irreducible affine toric varieties.

Since each cone in a fan is strongly convex, {0} ⊆ N is a face of all σ ∈ Σ. Hence we have T_{N} = Spec(ℂ[M]) ≃ (ℂ^{∗})^{n} ⊆ U_{σ} for all σ. These tori are all identified by the gluing, so we have T_{N} ⊆ X_{Σ}. We know from Chapter 1 that each U_{σ} has an action of T_{N}. The gluing isomorphism g_{σ_{1},σ_{2}} reduces to the identity mapping on ℂ[S_{σ_{1}∩σ_{2}}]. Hence the actions are compatible on the intersections of every pair of sets in the open affine cover, and patch together to give an action of T_{N} on X_{Σ}.

By Theorem 3.5 of Chapter 1, for each σ ⊆ Σ, U_{σ} is a normal affine variety. Hence the variety X_{Σ} is normal by Proposition 0.12.

To see that X_{Σ} is separated it will suffice to show that for each pair of cones σ_{1}, σ_{2} in Σ, writing τ = σ_{1} ∩ σ_{2}, the image of the diagonal map

Δ : U_{τ} → U_{σ_{1}} × U_{σ_{2}}

is Zariski closed (Exercise 1.2). But Δ comes from the ℂ-algebra homomorphism

Δ^{∗} : ℂ[S_{σ_{1}}] ⊗_{ℂ} ℂ[S_{σ_{2}}] ⟶ ℂ[S_{τ}]

defined by χ^{m} ⊗ χ^{n} ↦ χ^{m+n}. By Proposition 1.3, Δ^{∗} is surjective, which shows that

ℂ[S_{τ}] ≃ (ℂ[S_{σ_{1}}] ⊗_{ℂ} ℂ[S_{σ_{2}}])/I

for some ideal I. Hence the image of Δ is a Zariski closed subset of V_{σ_{1}} ×V_{σ_{2}}. □

Toric varieties were originally known as torus embeddings, and the variety X_{Σ} would be written T_{N}emb(Σ) in older references such as [26]. Other commonly used notations are X(Σ), or X(Δ), if the fan is denoted by Δ.

Every normal, separated toric variety is obtained as X_{Σ} for some fan Σ in N_{ℝ}. For example, Theorem 3.5 of Chapter 1 implies that every affine toric variety is X_{Σ} for the fan consisting of a single cone σ together with all of its faces. The construction of the projective toric variety associated to a lattice polytope in Chapter 2 shows that those varieties are also X_{Σ} for suitable fans Σ.

Proposition 1.6. Let P ⊂ M_{ℝ} ≃ ℝ^{n} be an n-dimensional lattice polytope. Then the projective toric variety X_{P} ≃ X_{Σ_{P}}, where Σ_{P} is the normal fan of P.

Proof. When P is very ample, this follows immediately from the description of the intersections of the affine open pieces of X_{P} in Proposition 3.12 from Chapter 2 and the definition of the normal fan Σ_{P}. The general case follows since the normal fans of P and kP are the same for all positive integers k. □

The general statement is a consequence of a theorem of Sumihiro from [34].

Theorem 1.7 (Sumihiro). Let the torus T_{N} act on an irreducible, normal, separated variety X. Then every point p ∈ X has a T_{N}-invariant affine open neighborhood. □

Corollary 1.8. Let X be an irreducible, normal, separated toric variety X containing the torus T_{N} as an affine open subset. Then there exists a fan Σ in N_{ℝ} such that X ≃ X_{Σ}.

The proof of Corollary 1.8, assuming Theorem 1.7, will be sketched in Exercise 2.8 in §2 after we have developed some of the properties of T_{N}-orbits on toric varieties.

Additional Examples. We now turn to some concrete examples. Many of these are toric varieties already encountered in previous chapters.

Example 1.9. Consider the fan Σ in N_{ℝ} ≃ ℝ^{2} in Figure 2. This is the same as the normal fan of each of the simplices P = kΔ_{2} as in Example 3.8 from Chapter 2. Here we show all points in the cones inside a rectangular viewing box (all figures of fans in the plane in this chapter will be drawn using the same convention.)

From the discussion in Chapter 2, we expect X_{Σ} ≃ ℙ^{2}, and we will show this in detail. The fan Σ has three two-dimensional cones σ_{0} = Cone(e_{1}, e_{2}), σ_{1} = Cone(−e_{1} − e_{2}, e_{2}), and σ_{2} = Cone(e_{1}, −e_{1} − e_{2}), together with the three rays τ_{ij} = σ_{i} ∩ σ_{j} for i ≠ j, and the origin. We see that the toric variety X_{Σ} is covered by the affine opens

U_{σ_{0}} = Spec(ℂ[S_{σ_{0}}]) ≃ Spec(ℂ[x, y]),
U_{σ_{1}} = Spec(ℂ[S_{σ_{1}}]) ≃ Spec(ℂ[x^{−1}, x^{−1}y]),
U_{σ_{2}} = Spec(ℂ[S_{σ_{2}}]) ≃ Spec(ℂ[xy^{−1}, y^{−1}]).

Figure 2. The fan Σ for ℙ^{2}

Moreover, by Proposition 1.3, the gluing data on the coordinate rings is given by the isomorphisms

g_{10}^{∗} : ℂ[x, y]_{x} ≃ ℂ[x^{−1}, x^{−1}y]_{x^{−1}},
g_{20}^{∗} : ℂ[x, y]_{y} ≃ ℂ[xy^{−1}, y^{−1}]_{y^{−1}},
g_{21}^{∗} : ℂ[x^{−1}, x^{−1}y]_{x^{−1}y} ≃ ℂ[x, xy^{−1}]_{xy^{−1}}.

It is easy to see that if we use the usual homogeneous coordinates (x_{0}, x_{1}, x_{2}) on ℙ^{2}, then x ↦ \frac{x_{1}}{x_{0}} and y ↦ \frac{x_{2}}{x_{0}} identifies the standard affine open U_{i} with U_{σ_{i}}. Hence we have recovered ℙ^{2} as the toric variety X_{Σ}. ⋄

Example 1.10. Generalizing Example 1.9, let N_{ℝ} ≃ ℝ^{n}, and write

e_{0} = −e_{1} − e_{2} − ⋯ − e_{n}.

Let Σ be the fan in N_{ℝ} consisting of the cones generated by all proper subsets of {e_{0}, …, e_{n}}. This is the normal fan of the n-simplex Δ_{n} and we know the toric variety X_{Σ} ≃ ℙ^{n} by Example 3.14 and Exercise 3.6 of Chapter 2. (You will check the details to verify that the general gluing construction gives the usual affine open cover of ℙ^{n} in Exercise 1.3). To relate this example to the discussion in Chapter 2, note that we have different embeddings of ℙ^{n} according to which kΔ_{n} is used to construct the Veronese mapping. However the normal fans of kΔ_{n} are all the same as this fan, and the varieties obtained this way for different k are isomorphic. ⋄

Example 1.11. Let N = N_{1} × N_{2} with (N_{1})_{ℝ} ≃ ℝ^{n} and (N_{2})_{ℝ} ≃ ℝ^{m}. Let Σ_{i} be the fan in (N_{i})_{ℝ} as in Example 1.10. Then Σ = Σ_{1} × Σ_{2} is a fan in N, consisting of all cones σ_{1} × σ_{2}, where σ_{i} is in Σ_{i}. By Example 4.8 of Chapter 2, the corresponding toric variety is the product ℙ^{n}×ℙ^{m}. For instance, if n = m = 1, we obtain the fan Σ ⊆ ℝ^{2} ≃ N_{ℝ} pictured in Figure 3.

Figure 3. A fan Σ with X_{Σ} ≃ ℙ^{1} × ℙ^{1}

This example was also studied in Example 4.10 in Chapter 2. Label the 2dimensional cones σ_{ij} = σ_{i} × σ′_{j} as above. Then

Spec(ℂ[S_{σ_{00}}]) ≃ ℂ[x, y],
Spec(ℂ[S_{σ_{10}}]) ≃ ℂ[x^{−1}, y],
Spec(ℂ[S_{σ_{11}}]) ≃ ℂ[x^{−1}, y^{−1}],
Spec(ℂ[S_{σ_{01}}]) ≃ ℂ[x, y^{−1}].

We see that if U_{0} and U_{1} are the standard affine open sets in ℙ^{1}, then U_{σ_{ij}} ≃ U_{i}×U_{j} and it is easy to check that the gluing makes X_{Σ} ≃ ℙ^{1} × ℙ^{1}. ⋄

Example 1.12. Let N = N_{1} × N_{2}, with N_{i} as in the previous example. Let Σ_{1} in (N_{1})_{ℝ} be as above, but let Σ_{2} be the fan consisting of the cone Cone(e_{1}, …, e_{n}) together with all its faces. Then Σ = Σ_{1} × Σ_{2} is a fan in N_{ℝ} and the the corresponding toric variety is X_{Σ} ≃ ℙ^{m} × ℂ^{n}. The case ℙ^{1} × ℂ^{2} was studied in Example 0.13. ⋄

Example 1.13. Let N ⊆ N′ be a sublattice. If Σ is a fan in N_{ℝ}, then we can also view it as a fan in (N′)_{ℝ}. What is the relation between X_{Σ,N} and X_{Σ,N′}? We will study this in much more detail in §3 of this Chapter. Here is a simple example showing one possibility. Let Σ be the fan pictured below, but viewed as a fan in (N′)_{ℝ} ≃ ℝ^{2}.

0 σ_{1}  σ_{0}

It is easy to check that the corresponding X_{Σ,N′} ≃ ℙ^{1} × ℂ^{∗} (Exercise 1.4). In general, when N is spanned by a subset of a basis for N′, we will show in §3 that X_{Σ,N′} ≃ X_{Σ} × T_{N′/N}. ⋄ Example 1.14. The two cones σ_{1} and σ_{2} in N_{ℝ} ≃ ℝ^{2} from Example 1.4 (see Figure 1), together with their faces, form a fan Σ. By comparing the descriptions of the coordinate rings of V_{σ_{i}} given there with what we did in Example 0.7, it is easy to check that X_{Σ} ≃ W, where W ⊆ ℙ^{1} × ℂ^{2} is the blowup of ℂ^{2} at the origin, defined as W = V(x_{0}y − x_{1}x) (Exercise 1.5).

Generalizing this, let N ≃ ℤ^{n}, let e_{1}, …, e_{n} ∈ N be the standard basis and let e_{0} = e_{1} + ⋯ + e_{n}. Let Σ be the fan in N_{ℝ} consisting of the cones generated by all subsets of {e_{0}, …, e_{n}} not containing {e_{1}, …, e_{n}}. Then the toric variety X_{Σ} is isomorphic to the blow-up of ℂ^{n} at the origin (see Exercise 0.8). ⋄

Example 1.15. Let r ∈ ℤ_{≥0} and consider the fan Σ in N_{ℝ} ≃ ℝ^{2} consisting of the four cones σ_{i} shown in Figure 4, together with all of their faces. The corresponding

Figure 4. A fan Σ with X_{Σ} ≃ H_{r}

variety X_{Σ} is covered by open affines, each isomorphic to ℂ^{2},

U_{σ_{1}} = Spec(ℂ[x, y])
U_{σ_{2}} = Spec(ℂ[x, y^{−1}])
U_{σ_{3}} = Spec(ℂ[x^{−1}, x^{−r}y^{−1}])
U_{σ_{4}} = Spec(ℂ[x^{−1}, x^{r}y]),

and glued according to the recipe from (1.2). The resulting X_{Σ} is called the Hirzebruch surface H_{r}. We saw a projective toric variety, the rational normal scroll S_{2,4}, defined by a polytope P with normal fan equal to Σ as here with r = 2 in Example 3.15 in Chapter 2. Note also that H_{0} ≃ ℙ^{1} × ℙ^{1}. We will study these varieties in detail in Chapter 8. ⋄

Example 1.16. Let q_{0}, …, q_{n} ∈ ℤ_{≥0} satisfy gcd(q_{0}, …, q_{1}) = 1. Consider the weighted projective space ℙ(q_{0}, …, q_{n}) introduced in Chapter 2. Let N = ℤ^{n+1}/ℤ · (q_{0}, …, q_{n}) and let v_{i}, i = 0, …, n be the images in N of the standard basis vectors in ℤ^{n+1}, so the relation

q_{0}v_{0} + ⋯ + q_{n}v_{n} = 0

holds in N. Let Σ be the fan made up of the cones generated by all the proper subsets of {v_{0}, …, v_{n}}. When q_{i} = 1 for all i, we obtain X_{Σ} ≃ ℙ^{n} by Example 1.10. And indeed, X_{Σ} ≃ ℙ(q_{0}, …, q_{n}) in general. This will be proved in Chapter 4 using a generalization of the homogeneous coordinates in ℙ^{n}.

Here, we will consider the special case ℙ(1, 1, 2), where v_{0} = −v_{1} − 2v_{2}. The fan Σ in N_{ℝ} is pictured in Figure 5 in the plane spanned by v_{1}, v_{2}. This

Figure 5. A fan Σ with X_{Σ} ≃ ℙ(1, 1, 2)

example has a different property from the others we have seen so far. Namely, consider the cone σ_{2} = Cone(v_{0}, v_{1}) = Cone(−v_{1} − 2v_{2}, v_{1}). We have σ_{2}^{∨} = Cone(−e_{2}, 2e_{1} − e_{2}) ⊆ M, so the situation is similar the case we studied in Example 2.19 in Chapter 1. Indeed, there is an change of coordinates defined by a matrix in GL(2, ℤ) that takes σ to the cone with d = 2 from that example. It follows that there is an isomorphism U_{σ_{2}} ≃ V(xz −y^{2}) ⊆ ℂ^{3} (Exercise 1.6). This is the rational normal cone Ĉ_{2}, hence has a singular point at the origin. The toric variety X_{Σ} is singular because of the singular point in this affine open subset. We also saw a polytope P such that the toric variety X_{P} ≃ ℙ(1, 1, 2) in Example 4.6 in Chapter 2, and the normal fan Σ_{P} coincides with the fan shown above. ⋄

The following theorem provides a dictionary between properties of X_{Σ} and properties of Σ that generalizes the results of Theorem 3.12 and Example 3.21 from Chapter 1. First, we introduce some terminology. The first two items here are parallel to those from Definition 2.14 from Chapter 1.

Definition 1.17. Let Σ ⊆ N_{ℝ} be a fan.

(a) We say Σ is smooth (or regular) if every cone σ in Σ is smooth (regular).

(b) We say Σ is simplicial if every cone σ in Σ is simplicial.

1. Fans and Normal Toric Varieties 109

(c) We say Σ is complete if its support |Σ| = ⋃_{σ∈Σ} σ is all of N_{ℝ}.

Theorem 1.18. Let X_{Σ} be the toric variety defined by a fan Σ ⊆ N_{ℝ}.

(a) X_{Σ} is a smooth variety if and only if the fan Σ is smooth.

(b) X_{Σ} is an orbifold (that is, X_{Σ} has only finite quotient singularities) if and only if the fan Σ is simplicial.

(c) X_{Σ} is compact in its classical topology if and only if Σ is complete.

Proof. Part a follows from the corresponding statement for affine toric varieties, Theorem 3.12, because smoothness is a local property (Definition 0.9). In part b, Example 3.21 from Chapter 1 gives one implication. The other will be proved in a future chapter. A proof of part c will be given in §4 of this chapter. □

For example, the blow-up of ℂ^{2} at the origin (Example 1.14) is not compact, since the support of the cones in the corresponding fan is not all of ℝ^{2}. The Hirzebruch surfaces H_{r} from Example 1.15 are smooth and compact because every cone in the corresponding fan is smooth, and the union of the cones is N_{ℝ} ≃ ℝ^{2}. The variety ℙ(1, 1, 2) from Example 1.16 is also compact, but it is not smooth. It is an orbifold (it has only finite quotient singularities) since the corresponding fan is simplicial.

Exercises for § 1.

1.1. Let Σ be a fan in N_{ℝ}. Show that the isomorphisms g_{σ_{1},σ_{2}} satisfy the compatibility conditions from §0 for gluing the V_{σ} along the subvarieties V_{σ_{1},σ_{2}}.

1.2. Let X be a variety obtained by gluing the affine open subsets {V_{α}} along open subsets V_{αβ} ⊆ V_{α} by isomorphisms g_{αβ} : V_{αβ} ≃ V_{βα}. Show that if the image of Δ : V_{αβ} → V_{α} × V_{β} defined by Δ(p) = (p, g_{αβ}(p)) is Zariski closed for all α, β, then X is separated.

1.3. Verify that if Σ is the fan given in Example 1.10, then X_{Σ} ≃ ℙ^{n}.

1.4. In Example 1.13, show that the toric variety X_{Σ,N′} ≃ ℙ^{1} × ℂ^{∗}.

1.5. Let N ≃ ℤ^{n}, let e_{1}, …, e_{n} ∈ N be the standard basis and let e_{0} = e_{1} + ⋯ + e_{n}. Let Σ be the set of cones generated by all subsets of {e_{0}, …, e_{n}} not containing {e_{1}, …, e_{n}}.

(a) Show that Σ is a fan in N_{ℝ}.

(b) Construct the affine open subsets covering the corresponding toric variety X_{Σ}, and give the gluing isomorphisms.

(c) Show that X_{Σ} is isomorphic to the blow-up of ℂ^{n} at the origin described in Exercise 0.8. Hint: The blow-up is the subvariety of ℙ^{n−1} × ℂ^{n} given by W = V(x_{i}y_{j} − x_{j}y_{i} | 1 ≤ i < j ≤ n). Cover W by affine open subsets W_{i} = W_{x_{i}} and compare those affines with your answer to part b.

1.6. In this exercise, you will verify claims made in Example 1.16.

(a) Show that there is a matrix A ∈ GL(2, ℤ) defining a change of coordinates that takes the cone in this example to the cone from Example 2.19 from Chapter 1, and find the mapping takes σ_{2}^{∨} to the dual cone.

(b) Show that Spec(ℂ[S_{σ_{2}}]) ≃ V(xz − y^{2}) ⊆ ℂ^{3}.

2. The Orbit–Cone Correspondence

In this section, we will study the orbits for the action of T_{N} on the toric variety X_{Σ}. Our main result will show that there is a bijective correspondence between the cones in Σ and the T_{N}-orbits. The connection comes ultimately from looking at limit points of one-parameter subgroups of T_{N} as in Chapter 1, (1.2).

A First Example. We introduce the key features by looking first at a concrete example.

Example 2.1. Consider ℙ^{2} ≃ X_{Σ} for the fan Σ from Figure 2 of §1. We have T_{N} ≃ (ℂ^{∗})^{2} ⊆ ℙ^{2} as the set of points with homogeneous coordinates (1, s, t), s, t ≠ 0. For each n = (a, b) ∈ N ≃ ℤ^{2}, we have the corresponding curve in ℙ^{2}:

λ^{n}(s) = (1, s^{a}, s^{b}).

(We are abusing notation slightly; strictly speaking according to the definition from Chapter 1, the one-parameter subgroup λ^{n} is a curve in (ℂ^{∗})^{2}, but we view it as a curve in ℙ^{2} via the inclusion (ℂ^{∗})^{2} ⊆ ℙ^{2}.) Since ℙ^{2} is compact (Theorem 1.18), lim_{s→0} λ^{n}(s) exists in ℙ^{2} for all n. We start by analyzing how the limit point depends on n. It is easy to check that the pattern is as follows.

n = (a, b)  lim_{s→0} λ^{n}(s)
a = b = 0  (1, 1, 1)
a = 0, b > 0  (1, 1, 0)
a > 0, b = 0  (1, 0, 1)
a = b < 0  (0, 1, 1)
a, b > 0  (1, 0, 0)
a < 0, b > a  (0, 1, 0)
b < 0, a > b  (0, 0, 1)

For instance, if a = b = −c < 0 for some c > 0 in ℤ, by the definition of the homogeneous coordinates in ℙ^{2}, (1, s^{−c}, s^{−c}) ∼ (s^{c}, 1, 1) and c > 0, so lim_{s→0}(s^{c}, 1, 1) = (0, 1, 1). You will check the remaining cases in Exercise 2.1.

Refering to Figure 2 from §1, note the correspondence between the regions in N described here and the cones in the fan Σ. In each case, we can describe the set of n giving one of the limit points as N ∩ relint(σ), where relint(σ) denotes the relative interior of one of the cones σ in Σ. In other words, we have recovered the structure of the fan Σ by considering these limits!

Now we relate this to the T_{N}-orbits in ℙ^{2}. By considering the description ℙ^{2} ≃ (ℂ^{3} \ {0})/ℂ^{∗}, you will see in Exercise 2.1 that there are exactly seven T_{N}-orbits in ℙ^{2}:

O_{1} = {(x_{0}, x_{1}, x_{2}) | x_{i} ≠ 0 for all i},
O_{2} = {(x_{0}, x_{1}, x_{2}) | x_{2} = 0, and x_{0}, x_{1} ≠ 0},
O_{3} = {(x_{0}, x_{1}, x_{2}) | x_{1} = 0, and x_{0}, x_{2} ≠ 0},
O_{4} = {(x_{0}, x_{1}, x_{2}) | x_{0} = 0, and x_{1}, x_{2} ≠ 0},
O_{5} = {(x_{0}, x_{1}, x_{2}) | x_{1} = x_{2} = 0, and x_{0} ≠ 0} = {(1, 0, 0)},
O_{6} = {(x_{0}, x_{1}, x_{2}) | x_{0} = x_{2} = 0, and x_{1} ≠ 0} = {(0, 1, 0)},
O_{7} = {(x_{0}, x_{1}, x_{2}) | x_{0} = x_{1} = 0, and x_{2} ≠ 0} = {(0, 0, 1)}.

The order these are listed shows the correspondence with the limit points above. In each case, note that the limit point lies in the corresponding orbit. Hence we obtain a correspondence between cones σ and orbits O by

σ ↔ O ⇔ lim_{s→0} λ^{n}(s) ∈ O for all n ∈ relint(σ).

These observations generalize to all toric varieties X_{Σ} and our goal in the rest of this section is to show this. ⋄

Points and Semigroup Homomorphisms. It will be convenient to use the intrinsic description of the points of an affine toric variety U_{σ} given in Proposition 3.1 from Chapter 1. We recall how this works and make some additional observations.

• Points of U_{σ} are in bijective correspondence with semigroup homomorphisms γ : S_{σ} → ℂ.

• For each cone σ we have a point of U_{σ} defined by

(2.1)  m ⟼ \begin{cases}1 & m ∈ S_{σ} ∩ σ^{⊥} \\ 0 & otherwise. \end{cases}

(This is a well-defined semigroup homomorphism since σ^{∨} ∩ σ^{⊥} is a face of σ^{∨}. If m, m′ ∈ S_{σ} and m + m′ ∈ S_{σ} ∩ σ^{⊥}, then both m, m′ ∈ S_{σ} ∩ σ^{⊥}.) We will denote this point by γ_{σ} and call it the distinguished point corresponding to σ.

• The point γ_{σ} is fixed under the T_{N}-action if and only if dim σ = dim N_{ℝ} (Corollary 3.3 from Chapter 1).

• If τ ⊆ σ is a face, then γ_{τ} ∈ U_{σ}. (This follows since σ^{⊥} ⊆ τ^{⊥}.)

Limits of One-Parameter Subgroups. It is not difficult to check that the limit points of one-parameter semigroups in Example 2.1 are exactly the distinguished points for the cones in the fan Σ for ℙ^{2} (Exercise 2.1). Our first general result will show that the same is true for all affine toric varieties U_{σ}.

Proposition 2.2. Let σ ⊆ N_{ℝ} be a strongly convex rational polyhedral cone and let n ∈ N. Then

n ∈ σ ⟺ lim_{s→0} λ^{n}(s) exists inU_{σ}.

Moreover, if n ∈ relint(σ), then lim_{s→0} λ^{n}(s) = γ_{σ}.

Proof. Let m ∈ S_{σ}. Then χ^{m}(λ^{n}(s)) = s^{⟨m,n⟩}. Hence lim_{s→0} χ^{m}(λ^{n}(s)) exists if and only if ⟨m, n⟩ ≥ 0. If n ∈ σ, then ⟨m, n⟩ ≥ 0 for all m ∈ S_{σ} = σ^{∨} ∩ M. This implies that m ↦ lim_{s→0} s^{⟨m,n⟩} is a semigroup homomorphism S_{σ} → ℂ, hence represents a point of U_{σ}. Conversely, if lim_{s→0} s^{⟨m,n⟩} exists in U_{σ}, then ⟨m, n⟩ ≥ 0 for all m ∈ σ^{∨}, so n ∈ (σ^{∨})^{∨} = σ. Finally, if n ∈ relint(σ), then ⟨m, n⟩ > 0 for all m ∈ S_{σ} \ σ^{⊥}, and ⟨m, n⟩ = 0 if m ∈ S_{σ} ∩ σ^{⊥}. Thus, the limit point is precisely the distinguished point γ_{σ} as above. □

Using this proposition, we can recover the fan Σ from X_{Σ} cone by cone as in Example 2.1. This is also the key observation needed for the proof of Corollary 1.8 from the previous section.

When the toric variety X_{Σ} is not compact, note that there will be some points n for which lim_{s→0} λ^{n}(s) does not exist. For instance, if X = ℂ^{2}, then the limit point lim_{s→0} λ^{n}(s) = lim_{s→0}(s^{a}, s^{b}) exists in X if and only if a, b ≥ 0. There are four different limit points and we obtain the subfan of Σ from Example 2.1 consisting of the cone σ_{0} from Figure 2 in §1, together with its two edges and the origin.

The Torus Orbits. Now we turn to the T_{N}-orbits in X_{Σ}. Fix one of the cones σ in Σ. From Exercise 3.1 of Chapter 1, we know that t ∈ T_{N} acts on the point represented by a semigroup homomorphism γ : S_{σ} → ℂ to yield the point represented by the semigroup homomorphism

(2.2)  t · γ : m ⟼ χ^{m}(t)γ(m).

From this description, it is easy to identify a certain subset of U_{σ} that is invariant under the T_{N}-action.

Lemma 2.3. Let σ be a strongly convex rational polyhedral cone, and define O(σ) to be the following set of semigroup homomorphisms representing points in a subset of U_{σ}:

O(σ) = {γ : S_{σ} → ℂ | γ(m) ≠ 0 for all m ∈ S_{σ} ∩ σ^{⊥},
γ(m) = 0 for all m ∈ S_{σ} \ σ^{⊥}}.

Then O(σ) is nonempty and invariant under the T_{N}-action on U_{σ}.

Proof. The distinguished point γ_{σ} lies in O(σ), so O(σ) is not empty. Invariance under T_{N} follows immediately from the definition and (2.2). □

In fact, we will show next that O(σ) is a T_{N}-orbit in U_{σ}. To see this we need to make several identifications.

First, notice that σ^{⊥} is the largest vector subspace of M_{ℝ} contained in σ^{∨}. Hence S_{σ} ∩ σ^{⊥} is actually a subgroup of S_{σ}. If γ(m) ≠ 0 for all m ∈ S_{σ} ∩ σ^{⊥}, then restricting γ to m ∈ S_{σ} ∩ σ^{⊥} = M ∩ σ^{⊥} yields a group homomorphism γ̂ : M ∩σ^{⊥} → ℂ^{∗} (see Exercise 2.2). Conversely, given any group homomorphism γ̂ : M ∩ σ^{⊥} → ℂ^{∗}, we obtain a semigroup homomorphism γ ∈ O(σ) by defining

γ(m) = \begin{cases}γ̂(m) & if m ∈ M ∩ σ^{⊥} \\ 0 & otherwise . \end{cases}

The argument sketched here can be completed to show that O(σ) has the structure of a group and that there is an isomorphism of groups

(2.3)  O(σ) ≃ Hom_{ℤ}(M ∩ σ^{⊥}, ℂ^{∗})

(you will check the details in Exercise 2.2).

Now recall Exercise 1.10 from Chapter 1, in which you showed that if T_{N} is a torus with character lattice M, then there is an isomorphism of groups

T_{N} ≃ Hom_{ℤ}(M, ℂ^{∗}).

The right side of (2.3) is analogous to the right side here, since M ∩ σ^{⊥} is a lattice in the subspace σ^{⊥} ⊆ M_{ℝ}. What plays the role of N, though? We claim that the answer is given as in the following Lemma.

Lemma 2.4. Let σ be the cone in N_{ℝ} in the above discussion. Let N_{σ} be the sublattice of N spanned by the points in σ ∩ N, and let N(σ) = N/N_{σ}. There is a dual pairing

⟨ , ⟩ : M ∩ σ^{⊥} × N(σ) → ℤ,

induced by the dual pairing ⟨ , ⟩ : M × N → ℤ. □

You will prove this in Exercise 2.3. The same reasoning as in Exercise 1.10 from Chapter 1 now shows that we have identified O(σ) as

(2.4)  O(σ) ≃ Hom_{ℤ}(M ∩ σ^{⊥}, ℂ^{∗}) ≃ T_{N(σ)}.

The exact sequence

(2.5)  0 ⟶ N_{σ} ⟶ N ⟶ N(σ) ⟶ 0

implies that T_{N(σ)} ≃ N(σ) ⊗_{ℤ} ℂ^{∗} is a quotient of T_{N} ≃ N ⊗_{ℤ} ℂ^{∗}. The action of T_{N} on this quotient torus is transitive, so O(σ) is a T_{N}-orbit in U_{σ}.

The Orbit-Cone Correspondence. Our next theorem is the major result of this section.

Theorem 2.5 (Orbit-Cone Correspondence). Let X_{Σ} be the toric variety associated to the fan Σ in N_{ℝ}.

(a) There is a bijective correspondence

{cones σ in Σ} ⟷ {T_{N} − orbits in X_{Σ}}
σ ⟷ O(σ) ≃ Hom_{ℤ}(M ∩ σ^{⊥}, ℂ^{∗}).

(b) Let k = dim_{ℝ} N_{ℝ}. For each cone σ, dim_{ℂ} O(σ) = k − dim_{ℝ} σ.

(c) O(σ) is the unique closed T_{N}-orbit in the affine open set U_{σ}, and

(2.6)  U_{σ} = ⋃_{{τ|τ is a face of σ}} O(τ).

(d) τ is a face of σ if and only if O(σ) ⊆ \overline{O(τ)}, and

(2.7)  \overline{O(τ)} = ⋃_{\{σ|τ is a face of σ\}} O(σ).

For instance, in Example 2.1, there are three types of cones and torus orbits in the variety ℙ^{2}.

• The cone ρ = {(0, 0, 0)} corresponds to the orbit O(ρ) = T_{N} ⊆ ℙ^{2}, which satisfies dim_{ℂ} O(ρ) = 2 = 2 − dim_{ℝ} ρ. This ρ is a face of all the other cones in Σ, and hence all the other orbits are contained in the closure of this one by part d. According to part c, since there are no cones properly contained in ρ, U_{ρ} = O(ρ) ≃ (ℂ^{∗})^{2}.

• The three one-dimensional cones τ give the torus orbits of dimension 1. Each of these is isomorphic to ℂ^{∗}. The closure of one of these orbits is one of the coordinate axes V(x_{i}) ⊆ ℙ^{2}, a copy of ℙ^{1}. Note that each τ is contained in two maximal cones.

• The three maximal cones σ_{i} in the fan Σ correspond to the three fixed points (1, 0, 0), (0, 1, 0), (0, 0, 1) of the torus action on ℙ^{2}. There are two of these in the closure of each of the 1-dimensional torus orbits.

Proof. Proposition 2.2 shows that the statement of part a is true for the affine toric variety U_{σ} and the faces τ ⊆ σ. We have

τ ↔ O = O(τ) ⇔ lim_{s→0} λ^{n}(s) = γ_{τ} ∈ O for all n ∈ relint(τ).

The variety X_{Σ} is obtained by gluing several of these affine opens, and part a follows in general from the fact that the gluing isomorphisms are T_{N}-equivariant. If τ = σ_{1} ∩ σ_{2}, then the orbits O(ρ) ⊆ U_{σ_{1}} and O(ρ) ⊆ U_{σ_{2}} for all faces ρ of τ are identified in the gluing. Part b follows from (2.4) and the following comments.

We will prove parts c and d together. We start with (2.6) from part c. If γ is a semigroup homomorphism γ : S_{σ} → ℂ representing a point p ∈ U_{σ}, then in Exercise 2.4, you will show that {m ∈ S_{σ} | γ(m) ≠ 0} = ρ ∩ M for some face ρ of σ^{∨}. By Proposition 2.9 from Chapter 1, ρ = σ^{∨} ∩ τ^{⊥} for some face τ of σ. Hence

{m ∈ S_{σ} | γ(m) ≠ 0} = S_{σ} ∩ τ^{⊥},

and the set of all γ which are nonzero precisely on this set is just the subset O(τ). This shows

U_{σ} ⊆ ⋃_{{τ|τ is a face of σ}} O(τ).

The opposite inclusion is clear, so (2.6) follows. Since the O(τ) are orbits for a group action, note that this is a disjoint union. Moreover, since U_{σ} is an affine toric variety, it is invariant under the T_{N}-action and the O(τ) for faces τ ⊆ σ are the only orbits that intersect U_{σ}.

Next, we turn to part d. If O(σ) ⊆ \overline{O(τ)}, then O(τ) ⊆ U_{σ}, hence τ is a face of σ by (2.6).

Conversely, if τ is a face of σ, then σ^{∨} ∩ τ^{⊥} is a face of σ^{∨}. We want to show that O(σ) ⊆ \overline{O(τ)}. To prove this, it suffices to show that \overline{O(τ)} ∩ O(σ) ≠ ∅ (Exercise 2.5). We will do this by using limits of one-parameter semigroups as in Proposition 2.2.

Let γ_{τ} be the semigroup homomorphism corresponding to the distinguished point of U_{τ}, so γ_{τ} (m) = 1 if m ∈ M ∩τ^{⊥}, and 0 otherwise. Let n ∈ relint(σ), and for each s ∈ ℂ^{∗} consider γ(s) = λ^{n}(s) · γ_{τ}. This is a semigroup homomorphism which maps

m ⟼ χ^{m}(λ^{n}(s))γ_{τ} (m) = s^{⟨m,n⟩}γ_{τ} (m).

For all s ∈ ℂ^{∗}, γ(s) ∈ O(τ) since the orbit of γ_{τ} is O(τ). Now let s → 0. Since n ∈ relint(σ), ⟨m, n⟩ > 0 if m ∈ σ^{∨} \ σ^{⊥}, and = 0 if m ∈ σ^{⊥}. It follows that γ(0) = lim_{s→0} γ(s) exists as a point in U_{σ} by Proposition 2.2, and represents a point in O(σ). But it is also in the closure of O(τ) by construction, so O(σ) ∩ \overline{O(τ)} ≠ ∅. This establishes the first part of d, and (2.7) follows from that.

Finally, the first part of c is a consequence of the rest (Exercise 2.6). □

In the example of ℙ^{2}, the orbit closures \overline{O(τ)} also have the structure of toric varieties. The same is true in general. The torus that acts is the quotient torus T_{N(τ)} from (2.3). The corresponding fan can be described as follows. Consider the set of cones σ in Σ containing τ as a face. For each such σ, let \overline{σ} be the image cone in N(τ)_{ℝ} under the quotient map

N_{ℝ} ⟶ N(τ)_{ℝ} ⟶ 0

obtained from (2.5). Then

(2.8)  Star(τ) = \{\overline{σ} ⊆ N(τ)_{ℝ} | τ is a face of σ\}

is a fan in N(τ)_{ℝ} (Exercise 2.9).

Corollary 2.6. For each cone τ in Σ, the orbit closure \overline{O(τ)} is isomorphic to the toric variety X_{Star(τ)}.

Proof. This follows from parts a and d of Theorem 2.5 (Exercise 2.9). □

The Variety V(xy − zw). We next turn to another example.

Example 2.7. Consider the affine toric variety X = V(xy − zw) studied in a number of examples from Chapter 1. For instance, in Example 1.17 from that Chapter, we showed that X is the normal toric variety corresponding to some cone σ. The dual cone is

(2.9)  σ^{∨} = Cone(e_{1}, e_{2}, e_{3}, e_{1} + e_{2} − e_{3}),

and X = Spec(ℂ[M ∩ σ^{∨}]).

Let us see how the results of this section apply to this example. In Example 1.17 from Chapter 1, we introduced the torus T = (ℂ^{∗})^{3} included in X as the image of

(2.10)  (t_{1}, t_{2}, t_{3}) ↦ (t_{1}, t_{2}, t_{3}, t_{1}t_{2}t_{3}^{−1}).

Given n = (a, b, c) ∈ N = ℤ^{3}, we have the one-parameter subgroup

(2.11)  λ^{n}(s) = (s^{a}, s^{b}, s^{c}, s^{a+b−c})

contained in X, and we proceed to examine limit points. Clearly, the limit point is in X if and only if a, b, c ≥ 0 and a + b ≥ c. These conditions determine the cone σ ⊆ N_{ℝ} given by

(2.12)  σ = Cone(e_{1}, e_{2}, e_{1} + e_{3}, e_{2} + e_{3}).

It is easy to check that this cone and the cone from (2.9)are in fact dual to each other (Exercise 2.10). ⋄

Final Comments. The technique of using limit points of one-parameter subgroups to study a group action is also a major tool in Geometric Invariant Theory as in [24], where the main problem is to construct varieties (or possibly more general objects) representing orbit spaces for the actions of algebraic groups on varieties. We will see how some of the ideas from that branch of algebraic geometry apply in the study of toric toric varieties in Chapter 4 and Chapter 10.

Exercises for § 2.

2.1. In this exercise, you will verify the claims made in Example 2.1 and the following discussion.

(a) Show that the remaining limits of one-parameter subgroups ℙ^{2} are as claimed in the example.

(b) Show that the (ℂ^{∗})^{2} orbits in ℙ^{2} are as claimed in the example.

2. The Orbit–Cone Correspondence 117

(c) Show that the limit point equals the distinguished point p_{σ} of the corresponding cone in each case.

2.2. In this exercise, you will supply some of the details that let us identify O(σ) from Lemma 2.3 as a T_{N}-orbit in U_{σ}.

(a) Show that if γ ∈ O(σ), then γ̂ : M ∩ σ^{⊥} → ℂ^{∗} is a group homomorphism.

(b) Deduce that O(σ) has the structure of a group.

(c) Verify carefully that we have an isomorphism of groups O(σ) ≃ Hom_{ℤ}(M ∩σ^{⊥}, ℂ^{∗}).

2.3. Prove Lemma 2.4

2.4. Show that if γ : S_{σ} → ℂ is a semigroup homomorphism, then {m ∈ S_{σ} | γ(m) ≠ 0} = ρ ∩ M for some face ρ of σ^{∨}.

2.5. Show that if τ, σ are two cones and \overline{O(τ)} ∩ O(σ) ≠ ∅, then O(σ) ⊆ \overline{O(τ)}. Hint: Show \overline{O(τ)} is invariant under the action of T_{N}.

2.6. Show that (2.6) and (2.7) imply the first part of c in Theorem 2.5.

2.7. In Proposition 3.16 of Chapter 1, we saw that if τ is a face of the strongly convex rational polyhedral cone σ in N_{ℝ} then U_{τ} = Spec(ℂ[S_{τ}]) is an affine open subset of U_{σ} = Spec(ℂ[S_{σ}]). In this exercise, we will show that the converse is also true. Namely we will see that if τ ⊆ σ and the induced map of affine toric varieties φ : U_{τ} → U_{σ} is an open immersion, then τ is a face of σ.

(a) Let n, n′ ∈ N ∩σ, and assume n+n′ ∈ τ. Show that lim_{s→0} λ^{n}(s)·lim_{s→0} λ^{n′}(s) ∈ U_{τ}.

(b) Show that lim_{s→0} λ^{n}(s) and lim_{s→0} λ^{n′}(s) are each in U_{τ}. Hint: use the description of points as semigroup homomorphisms.

(c) Deduce that n, n′ ∈ τ, so τ is a face of σ.

2.8. In this exercise, you will use Proposition 2.2 and Theorem 2.5 to deduce Corollary 1.8 from Theorem 1.7.

(a) By Theorem 1.7, and the results of Chapter 1, a separated toric variety has an open cover consisting of affine toric varieties U_{i} = U_{σ_{i}} for some collection of cones σ_{i}. Show that for all i, j, U_{i} ∩ U_{j} is also affine. Hint: Use the fact that X is separated.

(b) Show that U_{i} ∩ U_{j} is the affine toric variety corresponding to the cone τ = σ_{i} ∩ σ_{j}. Hint: Exercise 2.2 will be useful.

(c) If τ = σ_{i} ∩ σ_{j}, then show that τ is a face of both σ_{i} and σ_{j}. Hint: See Exercise 2.7 above.

(d) Deduce that X ≃ X_{Σ} for the fan consisting of the σ_{i} and all their faces.

2.9. Let τ be a cone in a fan Σ, and let Star(τ) be as defined in (2.8).

(a) Show that Star(τ) is a fan in N(τ)_{ℝ}.

(b) Prove Corollary 2.6.

2.10. In this exercise, you will check the claims made in Example 2.7.

(a) Show that the cones in (2.12) and (2.9) are dual.

(b) Identify the limits of one-parameter semigroups in this example, and describe the orbit-cone correspondence in this case.

(c) Show that the matrix

A = (\begin{matrix}1 & 1 & −1 \\ 1 & 0 & 0 \\ −1 & 0 & 1 \end{matrix})

defines an automorphism of N ≃ ℤ^{3} and the corresponding linear map on N_{ℝ} maps the cone σ^{∨} to σ.

(d) Deduce that the two affine toric varieties U_{σ} and U_{σ^{∨}} are isomorphic. Hint: Use Proposition 3.15 from Chapter 1.

3. Equivariant Maps of Toric Varieties

Recall from §0 of this chapter that if X and Y are varieties with affine open covers X = ⋃_{α} U_{α} and Y = ⋃_{β} U′_{β}, then a morphism of varieties φ : X → Y is a Zariski-continuous mapping such that the restrictions

φ|_{U_{α}∩φ^{−1}(U′_{β})} : U_{α} ∩ φ^{−1}(U′_{β}) ⟶ U′_{β}

are morphisms of affine varieties for all α, β.

When X and Y are normal toric varieties, the results on mappings of affine toric varieties from Proposition 3.14 and Proposition 3.15 from Chapter 1 yield a class of morphisms whose construction comes directly from the combinatorics of the associated fans. Our goal in this section is to study these special toric morphisms. We start by introducing some terminology.

Definition 3.1. Let N, N′ be two lattices, let Σ be a fan in N_{ℝ}, and let Σ′ be a fan in N′_{ℝ}. A ℤ-linear mapping \overline{φ} : N → N′ is compatible with the fans Σ and Σ′ if for every cone σ in Σ, there exists a cone σ′ in Σ′ such that \overline{φ}_{ℝ}(σ) ⊆ σ′.

Example 3.2. Let N ≃ ℤ^{2} with basis {e_{1}, e_{2}} and Σ be the fan from Figure 4 in § 1, so from Example 1.15, X_{Σ} is the Hirzebruch surface H_{r}. Let N′ ≃ ℤ with basis {f} and consider the fan Σ′

0 σ_{1}  σ_{0}

where σ_{1} = Cone(−f), and σ_{0} = Cone(f). X_{Σ′} ≃ ℙ^{1} (see Example 4.10 in Chapter 2). The mapping

\overline{φ} : N → N′, ae_{1} + be_{2} ↦ af

is compatible with the fans Σ and Σ′ since each cone in Σ maps onto one of the cones in Σ′. If r ≠ 0, on the other hand, the mapping

\overline{ψ} : N → N′, ae_{1} + be_{2} ↦ bf

is not compatible with these fans since the cone σ_{3} in Σ does not map into just one of the cones of Σ′. ⋄ Toric morphisms. It will be convenient in the following to take the result of Proposition 3.14, part a, from Chapter 1 as the definition of a toric morphism in this context. The discussion from Chapter 1 shows that this is consistent with what we did before for affine toric varieties.

Definition 3.3. Let X_{Σ}, X_{Σ′} be normal toric varieties, with Σ a fan in N and Σ′ a fan in N′. A morphism φ : X_{Σ} → X_{Σ′} is toric if φ maps the torus T_{N} ⊆ X_{Σ} into T_{N′} ⊆ X_{Σ′} and φ|_{T_{N}} is a group homomorphism.

The proof of part b of Proposition 3.14 from Chapter 1 generalizes immediately to show that any toric morphism φ : X_{Σ} → X_{Σ′} is an equivariant mapping for the T_{N}- and T_{N′}-actions. That is, we have a commutative diagram

(3.1)  \xymatrix{T_{N} × X_{Σ} \ar[r]^{Φ} \ar[d]_{φ|T_{N}×φ} & X_{Σ} \ar[d]^{φ} \\ T_{N′} × X_{Σ′} \ar[r]^{Φ′} & X_{Σ′}}

where Φ and Φ′ give the torus actions.

Our first result in this section shows that toric morphisms φ : X_{Σ} → X_{Σ′} are in bijective correspondence with ℤ-linear mappings \overline{φ} : N → N′ that are compatible with the fans Σ and Σ′.

Theorem 3.4. Let N, N′ be lattices, and let Σ in N and Σ′ in N′ be fans.

(a) If \overline{φ} : N → N′ is a ℤ-linear map that is compatible with Σ and Σ′, then there is a corresponding toric morphism φ : X_{Σ} → X_{Σ′} with the property that φ|_{T_{N}} is the map

\overline{φ} ⊗ 1 : N ⊗_{ℤ} ℂ → N′ ⊗_{ℤ} ℂ.

(b) Conversely, if φ : X_{Σ} → X_{Σ′} is a toric morphism, then φ induces a ℤ-linear map \overline{φ} : N → N′ that is compatible with the fans Σ and Σ′.

Proof. To prove part a, let σ be a cone in Σ. Because φ is compatible with the fans Σ and Σ′, there is some cone σ′ in Σ′ such that \overline{φ}_{ℝ}(σ) ⊆ σ′. Then Proposition 3.15 from Chapter 1 shows that \overline{φ} induces a toric morphism φ_{σ} : U_{σ} → U_{σ′}. Using the general criterion for gluing morphisms from Exercise 3.1, you will show in Exercise 3.2 that the φ_{σ} glue together to give a morphism φ : X_{Σ} → X_{Σ′}. Moreover, φ is toric because the ℤ-linear map \overline{φ} : N → N′ induces the group homomorphism φ : T_{N} → T_{N′} on T_{N} ⊆ U_{σ} for all σ by the comments after Proposition 3.14 in Chapter 1.

For part b, we show first that the toric morphism φ induces a ℤ-linear map \overline{φ} : N → N′. This follows since φ|_{T_{N}} is a group homomorphism. The lattices M, M′ are the lattices of characters of the two tori. For χ′ ∈ M′, the mapping χ′ ↦ χ′ ∘ φ|_{T_{N}} defines a group homomorphism M′ → M. Hence the dual of this map is the \overline{φ} : N → N′ we want.

It remains to show that \overline{φ} is compatible with the lattices Σ and Σ′. Because of the equivariance (3.1), each T_{N}-orbit O ⊆ X_{Σ} is mapped into a T_{N′}-orbit O′ ⊆ X_{Σ′}. By the Orbit-Cone Correspondence (Theorem 2.5, part a), each T_{N}-orbit is O = O(σ) for some cone σ in Σ, and similarly each T_{N′}-orbit is O′ = O(σ′) for some cone σ′ in Σ′. Furthermore, if τ ⊆ σ is a face, then by the same reasoning, there is some cone τ′ in Σ′ such that φ(O(τ)) ⊆ O(τ′).

We claim that in this situation τ′ must be a face of σ′. This follows since O(σ) ⊆ \overline{O(τ)} by part d of Theorem 2.5. Since φ is continuous in the classical topology, φ (\overline{O(τ)}) ⊆ \overline{O(τ′)}. But the only orbits contained in the closure of O(τ′) are the orbits corresponding to cones which have τ′ as a face. So τ′ is a face of σ′. It follows from part c of Theorem 2.5 that φ also maps the affine open subset U_{σ} ⊆ X_{Σ} into U_{σ′} ⊆ X_{Σ′}:

(3.2)  φ(U_{σ}) ⊆ U_{σ′}.

Now we use limit points of one-parameter semigroups once again to show that \overline{φ}_{ℝ}(σ) ⊆ σ′. By Proposition 2.2, if n ∈ N ∩ σ, then lim_{s→0} λ^{n}(s) exists in U_{σ}. By (3.2), φ(lim_{s→0} λ^{n}(s)) is a point of U_{σ′}. But by continuity of φ,

φ(lim_{s→0} λ^{n}(s)) = lim_{s→0} φ(λ^{n}(s))
= lim_{s→0} λ^{\overline{φ}(n)}(s).

Therefore, by Proposition 2.2 again, \overline{φ}(n) ∈ σ′. It follows easily that \overline{φ}_{ℝ}(σ) ⊆ σ′ since the cones are rational. Hence \overline{φ} is compatible with the fans Σ and Σ′. □

First examples of toric morphisms. Here are some first examples of toric morphisms defined by mappings compatible with the corresponding fans.

Example 3.5. Let N ≃ ℤ^{2}, N′ ≃ ℤ, and let

\overline{φ} : N → N′, ae_{1} + be_{2} ↦ af,

(the first mapping in Example 3.2). We saw that \overline{φ} is compatible with the fans Σ definining the Hirzebruch surface and Σ′ defining ℙ^{1}. Theorem 3.4 implies there is a corresponding toric morphism φ : H_{r} → ℙ^{1}. We will see later in this section that this mapping gives H_{r} the structure of a ℙ^{1}-bundle over ℙ^{1}. ⋄

Example 3.6. Let Σ be any fan in N ≃ ℤ^{n}. If k ∈ ℤ_{≥0}, then the multiplication map

\overline{φ} : N → N, n ↦ k · n

is compatible with Σ. Theorem 3.4 shows that there is a corresponding toric morphism φ : X_{Σ} → X_{Σ}, whose restriction to T_{N} ⊆ X_{Σ} is the group endomorphism

φ|_{T_{N}} (t_{1}, …, t_{n}) = (t_{1}^{k}, …, t_{n}^{k}).

For instance, if Σ is the fan in N ≃ ℤ^{2} from Figure 2 and k = 2, then we obtain the morphism φ : ℙ^{2} → ℙ^{2} defined in homogeneous coordinates by φ(x_{0}, x_{1}, x_{2}) = (x_{0}^{2}, x_{1}^{2}, x_{2}^{2}). ⋄

Change of lattice. Toric morphisms may also induced by the inclusion of one lattice in another. Let N′ be a lattice and let N ⊆ N′ be a sublattice. If Σ′ is a fan in whose cones are generated by elements of N, then we can view Σ′ as a fan either in N_{ℝ} or in N′_{ℝ}, and the inclusion map \overline{φ} : N → N′ is compatible with the fan Σ′ in the domain and codomain. As in Chapter 1, we obtain two different toric varieties X_{Σ′,N} and X_{Σ′,N′} depending on which lattice we consider, and the inclusion induces a toric morphism

φ : X_{Σ,N} → X_{Σ,N′}.

First let us consider the case where N is spanned by a subset of a basis for N′. We saw an instance of this in Example 1.13 from §1, and the general situation is very similar.

Proposition 3.7. Let N ⊆ N′ be a sublattice spanned by a subset of a basis for N′. Let Σ be a fan in N_{ℝ}. Then we have an isomorphism

φ : X_{Σ,N′} ≃ X_{Σ,N} × T_{N′/N},

where if dim N_{ℝ} = n and dim(N′)_{ℝ} = k, then T_{N′/N} ≃ (ℂ^{∗})^{n−k}.

Proof. This was proved for a fan consisting of a single cone σ and all of its faces in the course of the proof of Theorem 3.12 in Chapter 1 (see (3.2) in §3 of that chapter). The general case follows by the gluing construction. □

On the other hand, if N ⊆ N′ is a sublattice of finite index, then we have the following statement.

Proposition 3.8. Let N be a sublattice of finite index in N′, let Σ′ be a fan in N′_{ℝ} whose cones are generated by elements of N. Let G = N′/N. Then

φ : X_{Σ′,N} → X_{Σ′,N′}

induced by the inclusion φ : N → N′ presents X_{Σ′,N′} as the quotient X_{Σ′,N}/G.

Proof. By Proposition 3.18 of Chapter 1, since N has finite index in N′, the finite group G = N′/N acts on the affine open subset U_{σ′,N} ⊆ X_{Σ′,N} for each cone in Σ′. The coordinate ring of the affine open subset U_{σ′,N′} is the subring of Ginvariants in the coordinate ring of U_{σ′,N}, and hence U_{σ′,N′} is isomorphic to the quotient variety U_{σ′,N}/G. The actions by G on the affine opens are all compatible and glue together to give an action of G on X_{Σ′,N}. It follows that X_{Σ′,N′} is the quotient variety. □

Example 3.9. Let N′ ≃ ℤ^{2}, and Σ′ be the fan shown in Figure 5, so X_{Σ′,N′} is isomorphic to the weighted projective space ℙ(1, 1, 2). Let N be the sublattice of N′ given by N = {(a, b) ∈ N′ | b ≡ 0 mod 2}, so N′ has index 2 in N. This N′ is generated by v_{1} = e_{1}, v_{2} = 2e_{2}. Note that

v_{0} = −e_{1} − 2e_{2} = −v_{1} − v_{2} ∈ N.

Let \overline{φ} : N → N′ be the inclusion map. It is not difficult to see that with respect to the lattice N, X_{Σ′,N} ≃ ℙ^{2} (see Exercise 3.3). By Theorem 3.4, the ℤ-linear map \overline{φ} induces a toric morphism φ : ℙ^{2} → ℙ(1, 1, 2), and from Proposition 3.8, it follows that ℙ(1, 1, 2) ≃ ℙ^{2}/G.

Figure 6. The semigroups σ_{2}^{∨} ∩ M and σ_{2}^{∨} ∩ M′

It is instructive to consider how the dual cone σ_{2}^{∨} for σ_{2} in Figure 5 interacts with the lattice M dual to N. It is easy to check that M ≃ {(a, b/2) : a, b ∈ ℤ}, and σ_{2}^{∨} = Cone(2e_{1} − e_{2}, −e_{2}). In Figure 6, the points in σ_{2}^{∨} ∩ M′ are shown in white, and the points in σ_{2}^{∨} ∩ M not in σ_{2}^{∨} ∩ M′ are shown in black. Note that the picture in σ_{2}^{∨} ∩ M′ is the same (up to a change of coordinates in GL(2, ℤ)) as that shown in Figure 11 from Chapter 1. This shows again that ℙ(1, 1, 2) contains the affine open subset U_{σ_{2},N} isomorphic to the rational normal cone Ĉ_{2}. On the other hand U_{σ_{2},N′} ≃ ℂ^{2} is smooth. The other affine open subsets corresponding to σ_{1} and σ_{0} are isomorphic to ℂ^{2} in both ℙ^{2} and in ℙ(1, 1, 2). ⋄

In general, given any sublattice N ⊆ N′, a basis for N can be completed to a basis for a sublattice N′′ of finite index in N′. Given a fan Σ in N_{ℝ}, Propositions 3.7 and 3.8 show that the toric variety X_{Σ,N′} will be a quotient of the toric variety

X_{Σ,N′′} ≃ X_{Σ,N} × (ℂ^{∗})^{n−k}

by the finite abelian group N′/N′′. We leave it to the reader to construct examples of this general situation.

Refinements of fans and blow-ups. Next, let Σ, Σ′ be two fans in the same lattice N_{ℝ}, but assume Σ′ is a refinement of Σ (that is, for every cone σ′ in Σ′, there exists σ in Σ such that σ′ ⊆ σ. In this case, the identity mapping on N is automatically compatible with Σ and Σ′ and we obtain a toric morphism φ : X_{Σ′} → X_{Σ}.

Example 3.10. Consider the fan Σ′ in N ≃ ℤ^{2} pictured in Figure 1 fom §1 of this chapter. This is a refinement of the fan Σ consisting of Cone(e_{1}, e_{2}) and its faces. The corresponding toric varieties are X_{Σ} ≃ ℂ^{2} and X_{Σ′} ≃ W, the blow-up of ℂ^{2} at the origin (see Example 1.14). The toric morphism induced by the identity on N is φ : W → ℂ^{2}, the “blow-down” morphism that maps ℙ^{1} × {(0, 0)} ⊆ W to (0, 0) ∈ ℂ^{2} and is injective on the complement of ℙ^{1} × {(0, 0)} in W. ⋄

Figure 7. The star subdivision Σ^{∗}(σ)

We can generalize this example and Example 1.5 from §1 as follows.

Definition 3.11. Let Σ be a fan in N_{ℝ} where N ≃ ℤ^{n}. Let σ = Cone(v_{1}, …, v_{n}) be a smooth cone in Σ (that is, we require that {v_{1}, …, v_{n}} is a basis for ℤ^{n}). Let v_{0} = v_{1} + ⋯ + v_{n} and let Σ′(σ) be the fan consisting of all cones generated by subsets of {v_{0}, …, v_{n}} not containing {v_{1}, …, v_{n}}. Let

Σ^{∗}(σ) = (Σ \ {σ}) ∪ Σ′(σ).

Then Σ^{∗}(σ) is a fan in N_{ℝ}, called the star subdivision of Σ along σ.

Figure 7 shows the star subdivision of a three-dimensional cone σ = Cone(v_{1}, v_{2}, v_{3}) into three cones Cone(v_{0}, v_{1}, v_{2}), Cone(v_{0}, v_{1}, v_{3}), and Cone(v_{0}, v_{2}, v_{3}). The fan Σ^{∗}(σ) consists of these, together with their faces.

Proposition 3.12. With all notation as above, the toric variety X_{Σ^{∗}(σ)} is the blowup of X_{Σ} at the distinguished point γ_{σ} corresponding to the cone σ, and we have a toric morphism

φ : X_{Σ^{∗}(σ)} → X_{Σ}.

Proof. Since Σ and Σ^{∗}(σ) are the same outside the cone σ, without loss of generality, we may reduce to the case that Σ is the fan consisting of σ and all of its faces, and X_{Σ} is the affine toric variety U_{σ} ≃ ℂ^{n}.

Under the Orbit-Cone Correspondence (Theorem 2.5), σ corresponds to the distinguished point γ_{σ}, the origin (a fixed point for the torus action). By Theorem 3.4, the identity map on N induces a toric morphism

φ : X_{Σ^{∗}(σ)} → U_{σ} ≃ ℂ^{n}.

It is easy to check that the affine open sets covering X_{Σ^{∗}(σ)} are the same, and are glued together in the same way, as those for the blow-up of ℂ^{n} at the origin from Exercise 0.8. In Exercise 1.5 you will verify these claims. □

Even more generally, for any cone τ in Σ generated by a subset of a basis for the lattice N there exists a toric variety X_{Σ^{∗}(τ)} isomorphic to the blow-up of X_{Σ} along the orbit closure \overline{O(τ)} and a toric morphism

φ : X_{Σ^{∗}(τ)} → X_{Σ}.

The construction is a more or less direct generalization of the construction of Σ^{∗}(σ) as above; see [26, Proposition 1.26].

In Chapters 8 and 9, we will also see that toric blow-ups can be used to resolve singularities of toric varieties.

Exact Sequences and Fibrations. Next, we will consider some of the local structure of toric morphisms. To begin, consider a surjective ℤ-linear mapping

\overline{φ} : N ↠ N′.

If Σ in N_{ℝ} and Σ′ in (N′)_{ℝ} are compatible with φ, then we have a corresponding toric morphism

φ : X_{Σ} → X_{Σ′}.

It is easy to check that

Σ_{0} = \{σ ∈ Σ | σ ⊆ ker(\overline{φ})\}

is a fan in N_{ℝ} (a subfan of Σ), and that \overline{φ} is compatible with Σ_{0} and the trivial fan {0} in N′. This gives the toric morphism

φ|_{X_{Σ_{0}}} : X_{Σ_{0}} → T_{N′}.

In fact, by the reasoning used in the proof of Proposition 3.4, under φ : X_{Σ} → X_{Σ′},

(3.3)  φ^{−1}(T_{N′}) = X_{Σ_{0}}.

The cones from Σ_{0} also define a fan in the subspace (ker(φ))_{ℝ} ⊆ N_{ℝ}. Call this fan Σ′′. Moreover, it is always true that a basis for ker(φ) can be extended to a basis of N in this situation. By Proposition 3.7,

(3.4)  X_{Σ_{0}} ≃ X_{Σ′′} × (ℂ^{∗})^{n−k} ≃ X_{Σ′′} × T_{N′}.

Combining (3.3) and (3.4),

φ^{−1}(T_{N′}) ≃ X_{Σ′′} × T_{N′}.

In other words, over the affine open subset T_{N′} in X_{Σ′}, via the mapping φ, X_{Σ} is identified with the product of T_{N′} and the toric variety X_{Σ′′}. We say this subset of X_{Σ} is a fiber bundle over T_{N′} with fiber X_{Σ′′}. When the fan Σ has a suitable structure, we can make a similar statement globally.

Let N′′, N, N′ be lattices and let

(3.5)  0 ⟶ N′′ ⟶^{\overline{ψ}} N ⟶^{\overline{φ}} N′ ⟶ 0

be an exact sequence of ℤ-linear maps. Let Σ′′, Σ, and Σ′ be fans in (N′′)_{ℝ}, N_{ℝ}, and (N′)_{ℝ} respectively. Assume the mappings \overline{φ} and \overline{ψ} are compatible with the corresponding fans. We can view the cones in Σ′′ as cones in \overline{ψ}_{ℝ}((N′′)_{ℝ}) ⊆ N_{ℝ}.

Definition 3.13. In the situation of (3.5), we say Σ is split by Σ′ and Σ′′ if there exists a fan Σ̂′ in N_{ℝ} such that

(a) \overline{φ} maps each cone in Σ̂′ bijectively to a cone in Σ′, and

(b) each cone σ in Σ has the form σ = σ′ + σ′′ for some cone σ′ in Σ̂′ and σ′′ in Σ′′.

Theorem 3.14. Given an exact sequence (3.5), let Σ be a fan in N_{ℝ} split by fans Σ′ in (N′)_{ℝ} and Σ′′ in (N′′)_{ℝ}. Then X_{Σ} is a locally trival fiber bundle over X_{Σ′} with fiber X_{Σ′′}. That is, X_{Σ′} has a cover by open affine subsets U with the property that

φ^{−1}(U) ≃ U × X_{Σ′′}.

Proof. By condition a in Definition 3.13, we can identify Σ′′ with the set of cones in Σ that map to the trivial cone {0} in N′. As above, we have

φ^{−1}(T_{N′}) ≃ T_{N′} × X_{Σ′′}.

Then for all σ′ in Σ′, by condition b in Definition 3.13, we identify \{σ | \overline{φ}(σ) ⊆ σ′\} with

(3.6)  Σ(σ′) = {σ′ + σ′′ | σ′′ in Σ′′}.

This collection of cones in N also forms a fan, and the corresponding toric variety maps to U_{σ′} ⊆ X_{Σ′} under the restriction of the toric morphism φ. Hence because of the form of the cones in (3.6), it follows that

φ^{−1}(U_{σ′}) = X_{Σ(σ′)} ≃ U′_{σ} × X_{Σ′′},

and the theorem is proved. □

Example 3.15. To complete the discussion from Examples 3.2 and 3.5, consider the toric morphism φ : H_{r} → ℙ^{1} induced by the mapping

\overline{φ} : N → N′, ae_{1} + be_{2} ↦ af.

The fan Σ in N defining H_{r} is split by Σ′ and Σ′′ in this case because we have a fan Σ̂′ in N_{ℝ} consisting of

Cone(−e_{1} + re_{2}), {0}, Cone(e_{1}),

whose cones are mapped bijectively to the cones in Σ′ under φ_{ℝ}, and Σ′′ consisting of

Cone(e_{2}), {0}, Cone(−e_{2}).

Each cone in Σ has the form σ′+σ′′ for σ′ ∈ Σ̂′ and σ′′ ∈ Σ′′. Hence Theorem 3.14 shows that H_{r} is a locally trivial fibration over ℙ^{1}, with fibers isomorphic to

X_{Σ′′} ≃ ℙ^{1}.

If r > 0, then this fibration is not globally trivial. That is, it is not true that H_{r} ≃ ℙ^{1} × ℙ^{1}. There is some “twisting” on the fibers involved in the gluing of the U_{i} × ℙ^{1} in the construction of H_{r}. ⋄

We will give another, more precise, description of these fiber bundles and the “twisting” mentioned above using the language of sheaves in Chapter 6.

Exercises for § 3.

3.1. Let X be a variety with an affine open cover {U_{i}}, and let Y be a second variety. Let φ_{i} : U_{i} → Y be a collection of morphisms. We say that a morphism φ : X → Y is obtained by gluing the φ_{i} if φ|_{U_{i}} = φ_{i} for all i. Show that there exists such a φ : X → Y if and only if for every pair i, j,

φ_{i}|_{U_{i}∩U_{j}} = φ_{j}|_{U_{i}∩U_{j}}.

3.2. Let N, N′ be lattices, and let Σ in N_{ℝ} and Σ′ in (N′)_{ℝ} be fans. Let \overline{φ} : N → N′ be a ℤ-linear mapping that is compatible with the corresponding fans. Using Exercise 3.1 above, show that the induced toric morphisms φ_{σ} : U_{σ} → U′_{σ} glue together to form a morphism φ : X_{Σ} → X_{Σ′}.

3.3. This exercise asks you to verify some of the claims made in Example 3.9.

(a) Verify that with respect to the lattice N′, X_{Σ,N′} ≃ ℙ^{2}.

(b) Verify carefully that the affine open subset U_{σ_{2},N} ≃ Ĉ_{2}, the rational normal cone of degree 2.

3.4. Let Σ′ be the fan obtained from the fan Σ for ℙ^{2} in Example 1.9 by the following process. Subdivide the cone σ_{2} into two new cones σ_{21} and σ_{22} by inserting an edge Cone(−e_{2}).

(a) Construct an affine open cover for X_{Σ′} explicitly and give the gluing homomorphisms.

(b) Show that the resulting toric variety X_{Σ′} is smooth.

(c) Show that X_{Σ′} is isomorphic to the Hirzebruch surface H_{1}.

3.5. Let Σ′ be the fan obtained from the fan Σ for ℙ(1, 1, 2) in Example 1.16 by the following process. Subdivide the cone σ_{2} into two new cones σ_{21} and σ_{22} by inserting an edge Cone(−v_{2}).

(a) Construct an affine open cover for X_{Σ′} explicitly and give the gluing homomorphisms.

(b) Show that the resulting toric variety X_{Σ′} is smooth.

(c) Construct a morphism ϕ : X_{Σ′} → X_{Σ} and identify the fiber over the singular point of X_{Σ}.

(d) One of our smooth examples is isomorphic to X_{Σ′}. Which one is it?

4. Complete and Proper

The Compactness Criterion. In this section we will begin by proving part c of Theorem 1.18:

• The toric variety X_{Σ} is compact in its classical topology if and only if Σ is a complete fan (that is, |Σ| = ⋃_{σ∈Σ} σ = N_{ℝ}).

We will give a relatively elementary direct topological argument first, then reinterpret the key ideas to introduce a more general property of toric morphisms.

Proof. One part of this is an immediate consequence of Proposition 2.2 from §2, and we used this, for instance, in analyzing limit points of one-parameter subgroups in ℙ^{2} in Example 2.1 in that section. Namely, if X_{Σ} is compact, then lim_{s→0} λ^{n}(s) exists in X_{Σ} for every one-parameter subgroup λ^{n} of T_{N}, that is, for every n ∈ N. Because X_{Σ} is the union of the affine open subsets U_{σ} for σ in Σ, this implies

lim_{s→0} λ^{n}(s) ∈ U_{σ} for some σ.

But then Proposition 2.2 implies that n ∈ σ. Since every lattice point n ∈ N is contained in one of the cones of the fan, Σ is complete.

For the other implication, since X_{Σ} is separated (Theorem 1.5), X_{Σ} is Hausdorff as a topological space (Theorem 0.16). In fact more is true. The classical topology on each affine open set U_{σ} is even a metric topology. So compactness of X_{Σ} is equivalent to the Bolzano-Weierstrass property (the statement that every sequence of points in X_{Σ} has a convergent subsequence).

The proof will proceed by induction on d = dim_{ℝ} N_{ℝ}. In the case d = 1, the only complete fan Σ is the fan in ℝ pictured in Example 4.10 from Chapter 2, and in Example 1.13 of this chapter. The corresponding toric variety is X_{Σ} = ℙ^{1}. This is homeomorphic to S^{2}, the two-dimensional sphere, hence is compact.

Now, assume the statement has been proved for all complete fans Σ of dimension strictly less than d, and consider a complete fan Σ in N_{ℝ} ≃ ℝ^{d}. Let γ_{k} for k ∈ ℕ be a sequence of points in X_{Σ}. We will show that there is a convergent subsequence γ_{k_{ℓ}} in X_{Σ}.

Since Σ is consists of finitely many cones, without loss of generality we may assume the sequence γ_{k} lies entirely within one affine open subset U_{σ}. Let τ be a face of σ of dimension ≥ 1. By Corollary 2.6, the closure of the orbit O(τ) in X_{Σ} is the toric variety X_{Star(τ)} of dimension ≤ d − 1. If Σ is complete, then it is easy to check that the fan Star(τ) is also complete in N(τ)_{ℝ} (Exercise 4.2). If infinitely many terms in our sequence lie in \overline{O(τ)}, then the induction hypothesis implies that there is a convergent subsequence in \overline{O(τ)}. Hence, without loss of generality again, we may assume that our sequence lies entirely in the torus T_{N} ⊆ U_{σ}, with σ one of the maximal cones in Σ.

The key ingredient of the proof will be the following sort of logarithm mapping

(4.1)  L : T_{N} ≃ N ⊗_{ℤ} ℂ^{∗} ⟶ N_{ℝ}
n ⊗ s ⟼ n ⊗ log |s|

from T_{N} onto N_{ℝ}. For more properties of this mapping, see Exercise 4.3 below.

Since Σ is complete, the same is true for the fan consisting of the cones −σ, where σ is in Σ. Hence, by passing to a subsequence, we may assume that there is some σ′ in Σ such that

L(γ_{k}) ∈ −σ′

for all k ∈ ℕ. However, since the copies of the torus T_{N} in the U_{σ} for all the maximal cones in Σ are identified under the gluing in the construction of X_{Σ}, and the gluing map restricts to the identity on T_{N}, without loss of generality again, we can assume that σ = σ′ in the above, and

(4.2)  γ_{k} ∈ T_{N} ⊆ U_{σ} and L(γ_{k}) ∈ −σ

for all k ∈ ℕ. We must now interpret what this means in the intrinsic characterization of points of toric varieties used in §2 and §3 of this chapter.

Recall that points in T_{N} ≃ N ⊗_{Z} ℂ^{∗} can be thought of as group homomorphisms in Hom_{ℤ}(M, ℂ^{∗}) via the identification

n ⊗ s ⟷ (γ : m ↦ s^{⟨m,n⟩}).

Moreover, any such group homomorphism restricts to a semigroup homomorphism in Hom(M ∩ σ^{∨}, ℂ), hence yields a point γ in U_{σ}.

Write σ = Cone(v_{1}, …, v_{p}). Then for any point γ ↔ n ⊗ s ∈ T_{N} ⊆ U_{σ}, the condition L(γ) ∈ −σ as in (4.2) implies

n ⊗ log |s| = − ∑_{i=1}^{p} a_{i}v_{i} for some a_{i} ≥ 0.

Hence if m ∈ M ∩ σ^{∨},

log |γ(m)| = ⟨m, n⟩ log |s| = − ∑_{i=1}^{p} a_{i}⟨m, v_{i}⟩.

By definition of σ^{∨}, ⟨m, v_{i}⟩ ≥ 0 for all i. Since the a_{i} ≥ 0, this implies that

log |γ(m)| ≤ 0 ⟺ |γ(m)| ≤ 1 for all m ∈ M ∩ σ^{∨}.

We apply this to our sequence of points γ_{k}. Since the γ_{k} are a sequence of mappings to the closed unit disk in ℂ, which is compact, there is a subsequence γ_{k_{ℓ}} which converges to some γ ∈ Hom(M ∩ σ^{∨}, ℂ) = U_{σ}. You will check the details of this final assertion in Exercise 4.4. □

Proper Mappings. The property of compactness also has a relative version that is used most often in the theory of complex manifolds.

Definition 4.1. A continuous mapping f : X → Y is proper if the inverse image f^{−1}(T) is compact in X for every compact subset T ⊆ Y.

It is immediate that X is compact if and only if the constant mapping from X to the space Y = {pt} consisting of a single point is proper. This relative version of compactness may also be reformulated, for reasonably nice topological spaces, in the following way (see [14, Chapter 9, §4]).

Proposition 4.2. Let f : X → Y be a continuous mapping of locally compact Hausdorff spaces. Then f is proper if and only if f is a closed mapping (that is, f(U) is closed in Y for all closed subsets U ⊆ X), and all its fibers f^{−1}({y}), y ∈ Y, are compact. □

Although this criterion for properness can be used to characterize the proper toric morphisms, it is somewhat awkward because the fibers of general toric morphisms can be quite complicated. So we will look to another reformulation. Recall from §0 of this chapter that the fibers of a mapping can be described using the fibered product: if f : X → Y and y ∈ Y, then f^{−1}({y}) is X ×_{Y} {y} using the inclusion mapping {y} ↪ Y. In Exercise 4.5, you will show that for maps of locally compact Hausdorff spaces, the following universal closure property is equivalent to properness as in Definition 4.1.

Definition 4.3. A continuous mapping f : X → Y is said to be universally closed if for all spaces Z and all continuous mappings g : Z → Y, in the commutative diagram

\xymatrix{X ×_{Y} Z \ar[r]^{π_{X}} \ar[d]_{π_{Z}} & X \ar[d]^{f} \\ Z \ar[r]_{g} & Y}

the projection π_{Z} is a closed mapping.

Hence, in algebraic geometry, it is now customary to use the following definition of properness for morphisms of algebraic varieties.

Definition 4.4. A morphism of varieties φ : X → Y is proper if it is universally closed in the sense that for all varieties Z and morphisms ψ : Z → Y, in the commutative diagram

\xymatrix{X ×_{Y} Z \ar[r]^{π_{X}} \ar[d]_{π_{Z}} & X \ar[d]^{φ} \\ Z \ar[r]_{ψ} & Y}

the projection π_{Z} is a closed mapping in the Zariski topology. A variety X is said to be complete if the constant morphism φ : X → {pt} is proper.

Example 4.5. The Projective Extension Theorem [7, Chapter 8, §5, Theorem 6] shows that with X = ℙ^{n}, the mapping

π_{ℂ}^{m} : ℙ^{n} × ℂ^{m} → ℂ^{m}

is closed in the Zariski topology for all m. It follows that if V ⊆ ℂ^{m} is any affine variety, the projection

π_{V} : ℙ^{n} × V → V

is a closed mapping in the Zariski topology. By the gluing construction, the constant morphism ℙ^{n} → {pt} is proper, so ℙ^{n} is a complete variety (the prototypical complete variety). Moreover, any projective variety is complete (Exercise 4.6).

On the other hand consider the morphism ℂ → {pt}. We claim that this is not proper, so ℂ is not complete. To see this, consider ℂ ×_{{pt}} ℂ = ℂ^{2} and the diagram

\xymatrix{ℂ^{2} \ar[r]^{π_{2}} \ar[d]_{π_{1}} & ℂ \ar[d] \\ ℂ \ar[r] & \{pt\}}

The closed subset V(xy − 1) ⊆ ℂ^{2} does not map to a Zariski-closed subset of ℂ under π_{1}. Hence π_{1} is not a closed mapping, and ℂ is not complete. ⋄

Completeness is the algebraic version of compactness, and it can be shown that complete varieties are compact in the classical topology. Similarly, proper morphisms are proper maps as in Definition 4.1 when we consider the classical topology on the varieties X and Y.

The Properness Criterion. Part c of Theorem 1.18 can be understood as a special case of the following statement for toric morphisms.

Theorem 4.6. Let \overline{φ} : N → N′ be a ℤ-linear mapping compatible with fans Σ in N_{ℝ} and Σ′ ∈ (N′)_{ℝ}. The corresponding toric morphism

φ : X_{Σ} ⟶ X_{Σ′}

is proper if and only if

\overline{φ}_{ℝ}^{−1}(|Σ′|) = |Σ|.

Before we can prove this, however, we need to introduce yet another characterization of properness. While Definition 4.1 would lead us too far into the specifics of toric morphisms, Definition 4.4 is too general to be convenient as a way to prove a particular morphism is proper. As the second part of Example 4.5 indicates, in order for a morphism to be proper, we cannot have sequences of points on closed subsets of X ×_{Y} Z “escaping to infinity.” To test for that, it suffices to consider what happens on curves.

Theorem 4.7 (reference?). A morphism of varieties φ : X → Y is proper if and only if the following statement holds: If C is any smooth curve, p ∈ C is a point, and λ is any morphism

λ : C \ {p} ⟶ X

such that ψ = φ ∘ λ extends to a morphism \overline{ψ} : C → Y, then λ extends to a morphism \overline{λ} : C → X. In other words, if i is the inclusion C \ \{p\} ↪ C, then in any diagram

\xymatrix{C \ \{p\} \ar[r]^{λ} \ar[d]_{i} & X \ar[d]^{φ} \\ C \ar@{.>}[ur]_{\overline{λ}} \ar[r]_{\overline{ψ}} & Y}

where the outside square commutes, there exists an extension of λ to \overline{λ} : C → X making the entire diagram commute. □

Readers with some experience in the theory of schemes will recognize this statement as a version of the valuative criterion of properness ([17, Theorem 4.7 in Chapter II]). Although we have provided the foregoing remarks as context and motivation, for our purposes, nothing vital would be lost by simply taking this criterion as the definition of properness.

A smooth curve is a smooth variety of dimension 1 hence has the structure of a complex manifold of dimension 1 (a Riemann surface). By reducing to curves we can take advantage of the simple function theory on these varieties. In particular, at each point of a smooth curve there is a local analytic coordinate and we can represent functions by Laurent series. We are now ready to prove Theorem 4.6.

Proof. First we will show that if φ is proper, then \overline{φ}_{ℝ}^{−1}(|Σ′|) = |Σ|. By the definition of compatibility of \overline{φ} with the fans, the inclusion

\overline{φ}_{ℝ}^{−1}(|Σ′|) ⊇ |Σ|

is automatic. Because all the cones are rational, it will suffice consider n ∈ \overline{φ}_{ℝ}^{−1}(|Σ′|) ∩ N to show the opposite inclusion. Let \overline{φ}(n) = n′ ∈ σ′ in Σ′, and consider the one-parameter subgroup

λ^{n} : ℂ^{∗} → T_{N} ⊆ X_{Σ}.

This is a mapping from the curve ℂ^{∗} = ℂ \ {0} into X_{Σ}. We have

φ(λ^{n}(s)) = λ^{\overline{φ}(n)}(s) = λ^{n′}(s)

for all s ≠ 0.

Since n′ ∈ σ′, by Proposition 2.2, lim_{s→0} λ^{n′}(s) exists in U_{σ′} ⊆ X_{Σ′}. This means that λ^{n′} extends to a morphism

\overline{λ^{n′}} : ℂ → X_{Σ′},

and we have a commutative diagram as in the statement of Theorem 4.7:

\xymatrix{ℂ \ \{0\} \ar[r]^{λ^{n}} \ar[d]_{i} & X_{Σ} \ar[d]^{φ} \\ ℂ \ar@{.>}[ur]_{\overline{λ^{n}}} \ar[r]_{\overline{λ^{n′}}} & X_{Σ′} .}

Since we assume φ is proper, there exists an extension of λ^{n}, \overline{λ^{n}} : ℂ → X_{Σ} making the diagram commute. It follows that lim_{s→0} λ^{n}(s) exists and equals a point in X_{Σ}. Hence n ∈ σ for some σ in Σ by Proposition 2.2.

For the opposite implication, assume that \overline{φ}_{ℝ}^{−1}(|Σ′|) = |Σ|, and assume we have a commutative diagram

(4.3)  \xymatrix{C \ \{p\} \ar[r]^{λ} \ar[d]_{i} & X_{Σ} \ar[d]^{φ} \\ C \ar[r]_{\overline{ψ}} & X_{Σ′}}

where C is any smooth curve and p is a point of C. We must show that λ extends to \overline{λ} : C → X_{Σ} making the diagram commute. Let σ′ be a cone in Σ′ such that \overline{ψ}(p) ∈ U_{σ′}. Without loss of generality, we can replace X_{Σ′} by the affine toric variety U_{σ′}, and we can also take C affine such that λ(C \ {p}) ⊆ T_{N} and replace X_{Σ} by T_{N}. We want to see that there exists some cone σ in Σ such that we can define λ(p) ∈ U_{σ} to extend λ to C.

By our reductions, corresponding to the morphisms in (4.3), we have a commutative diagram of ℂ-algebra homomorphisms

(4.4)  \xymatrix{ℂ[C \ \{p\}] & ℂ[M] \ar[l]_{λ^{∗}} \\ ℂ[C] \ar[u]_{i^{∗}} & ℂ[(σ′)^{∨} ∩ M′] \ar[l]^{_{\overline{ψ}_{∗}}} \ar[u]_{φ^{∗}}}

The fact that ψ = φ ∘ λ extends to C means that every element of the image of \overline{ψ} ^{∗} is a function on C with no pole at p. We may use Laurent expansions in terms of a local analytic coordinate at p on C and write ord_{p} for the order of pole or vanishing. Then ord_{p}∘\overline{ψ} ^{∗} gives a ℤ-linear mapping from \{χ^{m′} : m′ ∈ M′\} ≃ M′ to ℤ that is nonnegative on (σ′)^{∨} ∩ M′. By duality, this is the same as an element of Hom_{ℤ}(M′, ℤ) = N′ that lies in (σ′)^{∨∨} = σ′.

Since \overline{φ}_{ℝ}^{−1}(|Σ′|) = |Σ|, there exists σ in Σ such that \overline{φ}_{ℝ}(σ) ⊆ σ′. Then the commutativity of (4.4)implies that ord_{p} ∘λ^{∗} ≥ 0 for all functions in ℂ[σ^{∨} ∩M] ⊆ ℂ[M] also. This gives the existence of the extension of λ, and hence φ is a proper morphism by Theorem 4.7. □

Example 4.8. An important class of examples of proper morphisms are the blowups

φ : X_{Σ^{∗}(σ)} → X_{Σ}

studied in Proposition 3.12. In that case, the hypotheses for Theorem 4.6 are satisfied since |Σ| = |Σ^{∗}(σ)| (the fan Σ^{∗}(σ) is a refinement of Σ with the same support), and \overline{φ} is the identity on N. We can also see that the resulting toric morphism is proper. For instance, in the blowup

φ : Bl_{0}(ℂ^{2}) → ℂ^{2},

the inverse image of each point other than (0, 0) ∈ ℂ^{2} is a single point, while the inverse image of (0, 0) is the compact set ℙ^{1} × (0, 0) ⊂ Bl_{0}(ℂ^{2}). In this case, it is not difficult to check directly that φ^{−1}(T) is compact for all compact T in ℂ^{2}. ⋄

Exercises for § 4.

4.1. Give a second argument for the implication

X_{Σ} compact ⇒ Σ complete

from part c of Theorem 1.18 using induction on the dimension d of N. Hint: If Σ is not complete and d > 1, then there is a one-dimensional cone τ in the boundary of the support of Σ. Consider the fan Star(τ) and the corresponding toric subvariety of X_{Σ}.

4.2. Let Σ be a complete fan in N_{ℝ}. Let τ be any cone in Σ and let Star(τ) be as in (2.8). Show that Star(τ) is a complete fan in N(τ)_{ℝ}.

4.3. In this exercise, you will develop some additional properties of the logarithm mapping from (4.1).

(a) Let S^{1} be the unit circle in the complex plane, a subgroup of the multiplicative group ℂ^{∗}. Show that there is an isomorphism of groups

φ : ℂ^{∗} ⟶ S^{1} × ℝ
z ⟼ (|z|, log |z|),

where the operation in the second factor on the right is addition.

(b) Show that the compact real n-dimensional torus (S^{1})^{n} can be viewed as a subgroup of T_{N}, and that T_{N}/(S^{1})^{n} ≃ N_{ℝ}. Hint: Use φ from part a.

(c) Let Σ be a fan in N. Show that the action of the compact real torus (S^{1})^{n} ⊆ T_{N} on T_{N} extends to an action on the toric variety X_{Σ}, and that the quotient space

(X_{Σ})/(S^{1})^{n} ≅ ⋃_{σ} N(σ)_{ℝ},

where ≅ denotes homeomorphism of topological spaces, and the union is over all cones in the fan. Hint: Use the Orbit-Cone Correspondence (Theorem 2.5).

(d) Let Σ in ℝ^{2} be the fan from Example 1.9, so X_{Σ} ≃ ℙ^{2}. Show that under the action of (S^{1})^{2} ⊆ (ℂ^{∗})^{2} as in part c, ℙ^{2}/(S^{1})^{2} ≅ Δ_{2}, the 2-dimensional simplex.

We will say much more about the topology of toric varieties in Chapter 12.

4.4. In this exercise, you will supply the details to complete the proof of part c of Theorem 1.18 from the start of this section. Assume that γ_{k} ∈ Hom(M ∩ σ^{∨}, ℂ) is a sequence of semigroup homomorphisms, and assume that |γ_{k}(m)| ≤ 1 for all m ∈ M ∩ σ^{∨} and all k ∈ ℕ. We want to show that there is a subsequence γ_{k_{ℓ}} that converges to some γ ∈ Hom(M ∩ σ^{∨}, ℂ).

(a) The semigroup S_{σ} = M ∩ σ^{∨} is generated by a finite Hilbert basis {m_{1}, …, m_{s}}. Use this fact and the compactness of {z ∈ ℂ | |z| ≤ 1} to show that there exists a subsequence γ_{k_{ℓ}} such that the sequences γ_{k_{ℓ}}(m_{j}) converge in ℂ for all j.

(b) Deduce that the subsequence γ_{k_{ℓ}} converges to a γ ∈ Hom(M ∩ σ^{∨}, ℂ).

4.5. Show that a mapping of locally compact topological spaces f : X → Y is proper if and only if it universally closed. Hint: Show first that compactness of X is equivalent to the statement that the mapping f : X → {pt} is universally closed. Then use the easy fact that any composition of universally closed mappings is universally closed.

4.6. Show that any projective variety is complete according to Definition 4.4.

Homogeneous Coordinates

0. Background: Quotients in Algebraic Geometry

• ℙ^{n} = ℂ^{n+1} \ 0/C^{∗}

• G ⊆ (C^{∗})^{n} closed subgroup (possibly disconnected, with torsion)

• Good categorical quotient(GCQ), Good geometric quotient (GGQ)

• Spec(A) = X → X/G = Spec(A^{G}) is GCQ

• A normal implies A^{G} normal

• ϕ(x) = ϕ(y) iff \overline{Gx} ∩ \overline{Gy} ≠ ∅

• X → Y normal, onto, π^{−1}(y) = orbit∀y implies GGQ [dolgachev p.95]

• V (xy − zw) GCQ but not GGQ

1. Total Coordinate Rings of Toric Varieties

• define Cl(X)

• M → ℤ^{Σ(1)} → Cl(X) → 0

• S = ℂ[X_{ρ} | ρ ∈ Σ(1)], grading

• Examples: ℙ^{n}, weighted projective space, ℙ^{n} × ℙ^{m}, Bl_{0}(C^{m}), Hirzebruchs • V(xy − zw), mirror symmetry (MS)

2. Quotient Constructions of Toric Varieties

• Assume Σ(1) spans N_{ℝ}

• 0 → M → ℤ^{Σ(1)} → Cl(X) → 0

• G = Hom_{ℤ}(Cl(X), ℂ^{∗})

• concrete description of action on ℂ^{Σ(1)}

136 4. Homogeneous Coordinates

• B = ⟨Xσ̂ | σ ∈ Σ_{max}⟩, example of ℙ^{n} with B = ⟨x_{0}, …, x_{n}⟩, G = ℂ^{∗} acts as usual

• ℂ^{Σ(1)} \ V (B)/G = X_{Σ} GCQ, GGQ iff Σ simplicial – maybe use Batyrev-Cox proof

3. Examples

• Examples from section 1

• MS case

• relate to change of lattice

4. Polytopes and Homogenization

• P corresponds to a degree in Cl(X), so S_{m deg} ↔ |mP ∩ M|

• pictures similar to “What is paper” of how to homogenize

• ℂ^{Σ(1)} \ V (B)/G ↔ ℂ^{∗} → ℙ^{n}

• Mirror constructions

Part II

Further Theory of Toric Varieties

Chapters 5 to 9 continue our development of the theory of toric varieties. This part of the book assumes more algebraic geometry than before. The chapters no longer have background sections; rather, the required algebraic geometry is developed as needed, with frequent references to basic texts such as [17] or [30]. As we will see, toric varieties provide a wonderful context in which to develop the basic tools of modern algebraic geometry.

Divisors on Toric Varieties

1. Normality and Divisors

• codimension one is locally principle

• DVR’s, v_{D}(f)

• Toric divisors, D = D_{ρ} then v_{D}(χ^{u}) = ⟨u, n_{ρ}⟩

• example of locally principal but not globally principal

• support of a divisor

2. Cartier and Weil Divisors on Toric Varieties

• Define O_{Y} (U) (don’t call it a sheaf)

• Affine torics Cartier iff principal, general condition ∑ a_{i}D_{i} Cartier iff ∀ maximal σ, ∃m_{σ} ∈ M such that a_{i} = ⟨m_{σ}, n⟩, ∀n ∈ σ

• T -invariant divisors

• X smooth implies Weil = Cartier; Cartier = Weil in general implies UFD but for torics implies smooth

3. Divisor Classes on Toric Varieties

• Class group

• Picard group

• Proof Cl(X) is same as that defined in Chapter 4

Sheaves on Toric Varieties

1. Invertible and Reflexive Sheaves

• O_{X}, sheaf of a Weil divisor O_{X}(D)

• formal definition of sheaf

• invertible sheaves (when is O_{X}(D) invertible)

• sheaf of modules (in general, O_{X}(D) need not be locally free, define reflexive)

• Invertible iff Cartier, every invertible is sheaf of a Cartier divisor

• linear equivalence of divisors corresponds to iso of sheaves (rmk: in appendix, defns of ⊗, reflexive module)

2. Connection to Polytopes

• Global sections, definition of P_{D}

• maps to ℙ^{n}

• support functions

• S_{α} = H^{0}(O_{X}(D)) ≃ ⊕_{m∈P_{D}}χ^{m}

• general defs for v.ample, ample, gen’d by sections

3. Ampleness Criteria on Toric Varieties

• toric criteria for nef, ample (example of not nef divisor on Hirzebruch

• Smooth then ample iff v.amp?

• toric Nakai

• nef cone, Mori cone

• primitive collections

• nef and not big implies mod by subspace get smaller fan

142 6. Sheaves on Toric Varieties

4. The Canonical Sheaf

• Ω^{1}, K, Gorenstein, Fano

• Cotangent s.e.s, residue sequence

• Graded S-module gives sheaf on X_{Σ}; examples for Ω_{X}, T_{X}

Sheaf Cohomology of Toric Varieties

1. Čech Cohomology

• general definition

• H^{i}(O(D)), semiampleness • H^{i}(Ω(D))

2. Vanishing Theorems

3. Serre Duality and Ehrhart polynomials

Toric Surfaces

1. Singularities of Toric Surfaces and Their Resolutions

In this chapter, we will apply most of the theory developed so far to give a detailed treatment of the structure of normal toric varieties of dimension two (toric surfaces).

Singular Points of Toric Surfaces. To begin, we note the following consequences of the construction of normal toric varieties and the correspondence between affine toric varieties and cones. If X_{Σ} is the toric surface corresponding to a fan Σ in N_{ℝ} ≃ ℝ^{2}, then the ray generator of each of the one-dimensional cones in Σ is a primitive vector (that is, it can be extended to a basis of N). It follows from Theorem 1.18 of Chapter 3 that the toric surface obtained by removing the fixed points of the torus action (that is, the points corresponding to the two-dimensional cones under the Orbit-Cone Correspondence) is smooth. There are only finitely many such points, so X_{Σ} contains at most a finite number of singular points. Moreover, two-dimensional cones are always simplicial, so from Example 3.21 in Chapter 1, each of these singular points is a finite abelian quotient singularity (isomorphic to the image of the origin in the quotient ℂ^{2}/G where G is a finite abelian group).

The following a normal form for two-dimensional strongly convex rational polyhedral cones in ℝ^{2} will facilitate the description of the singularities of toric surfaces.

Proposition 1.1. Let σ ⊆ N_{ℝ} ≃ ℝ^{2} be a two-dimensional strongly convex rational polyhedral cone in ℝ^{2}. Then there exists a basis {e_{1}, e_{2}} for N such that

σ = Cone(e_{2}, me_{1} − ke_{2}),

where m > 0, 0 ≤ k < m, and gcd(m, k) = 1.

Proof. We will need the following statement about integer arithmetic here and at several other points in this chapter.

Lemma 1.2. Let m > 0 and l be integers. There exist unique integers s and k such that

l = sm − k, 0 ≤ k < m.

The proof of the lemma is left to the reader as Exercise 1.1.

Say σ = Cone(v_{1}, v_{2}) where v_{i} are primitive vectors. Since v_{1} is primitive, we can take it as part of a basis of N, and we let e_{2} = v_{1}. Since σ is strongly convex, for any basis {e′_{1}, e_{2}} for N, it will be true that

v_{2} = me′_{1} + le_{2}

for some m ≠ 0. By replacing e′_{1} by −e′_{1} if necessary, we can assume m > 0. By Lemma 1.2, there are integers s, k such that l = sm − k, where 0 ≤ k < m. Using this integer s, let e_{1} = e′_{1} + se_{2}. Then {e_{1}, e_{2}} is also a basis for N and

v_{2} = me_{1} + (l − sm)e_{2} = me_{1} − ke_{2}.

Hence σ = Cone(e_{2}, me_{1} −ke_{2}) as claimed. The equation gcd(m, k) = 1 follows since v_{2} is primitive. The uniqueness of m, k is left to the reader as Exercise 1.2.

□

We will call the integers m, k in this statement the parameters of the cone σ, and {e_{1}, e_{2}} is called a normalized basis for N relative to σ. Using the normal form, we will next describe the local structure of the point p_{σ} in the affine toric variety U_{σ}. Recall from Example 3.21 in Chapter 1 that if N′ ⊆ N is the sublattice generated by the ray generators of σ, then U_{σ} ≃ ℂ^{2}/G, where G = N/N′. In our situation, N = ℤe_{1} ⊕ ℤe_{2} and

N′ = ℤe_{2} ⊕ ℤ(me_{1} − ke_{2}) = mℤe_{1} ⊕ ℤe_{2},

so it follows easily that

(1.1)  G = N/N′ ≃ ℤ/mℤ.

In particular, for singularities of toric surfaces, the finite group G is always cyclic. The action of G on ℂ^{2} is determined by the integers m, k as follows. We write

μ_{m} = {ζ ∈ ℂ : ζ^{m} = 1},

the group of mth roots of unity in ℂ. Then a choice of a primitive mth root of unity defines an isomorphism of groups μ_{m} ≃ ℤ/mℤ.

Proposition 1.3. Let M′ be the dual lattice of N′ and let u_{1}, u_{2} ∈ M′ be dual to v_{1}, v_{2} in N′. Using the coordinates x = χ^{u_{1}} and y = χ^{u_{2}} on ℂ^{2}, the action of ζ ∈ μ_{m} ≃ N/N′ is given by

ζ · (x, y) = (ζx, ζ^{k}y).

Proof. From the general discussion in Chapter 1, N/N′ acts on the coordinate ring of ℂ^{2} via

(1.2)  (v + N′) · χ^{u′} = e^{2πi⟨u′,v⟩}χ^{u′},

where u′ ∈ σ^{∨} ∩ M′ and each element of N/N′ is given by v = je_{1} for some j, 0 ≤ j ≤ m − 1.

An easy calculation shows ⟨u_{1}, e_{1}⟩ = 1/m and ⟨u_{2}, e_{1}⟩ = k/m. Hence if we set up the isomorphism μ_{m} ≃ N/N′ by mapping e^{2πij/m} ↦ je_{1} + N′, then for all ζ = e^{2πij/m} ∈ μ_{m}, from (1.2)

ζ · (x, y) = (e^{2πij/m}x, e^{2πijk/m}y) = (ζx, ζ^{k}y),

which is what we wanted to show. □

The isomorphism in (1.1) and the result of Proposition 1.3 also indicate that there is a slight but manageable ambiguity in the normal form for two-dimensional cones. We say cones are lattice equivalent if there is a bijective ℤ-linear mapping φ : N → N taking one cone to the other. After choice of basis for N, such mappings are defined by matrices in GL(2, ℤ).

Proposition 1.4. Let σ = Cone(e_{2}, me_{1} − ke_{2}) and σ̃ = Cone(e′_{2}, m̃e′_{1} − k̃e′_{2}) be cones in normal form that are lattice equivalent. Then m̃ = m and either k̃ = k or kk̃ ≡ 1 mod m.

Proof. Since the cones are lattice equivalent, writing N′ and Ñ′ for the sublattices as in (1.1), there is a bijective ℤ-linear mapping φ : N → N such that φ(N′) = Ñ′. Hence N/Ñ′ ≃ N/N′, so m̃ = m. The statement about k and k̃ is left to the reader in Exercise 1.2. □

We will next consider two examples to illustrate Proposition 1.3 and we will identify the resulting toric surface singularities.

Example 1.5. First consider a cone

σ = Cone(e_{2}, me_{1} − e_{2})

with parameters m > 1 (so the cone is not smooth) and k = 1. Apart from slightly different notation, this is precisely the cone considered in Example 2.19 from Chapter 1. The corresponding toric surface U_{σ} is the cone over the rational normal curve of degree m: Ĉ_{m} ⊆ ℂ^{m+1}. The quotient structure was studied in the special case m = 2 in Example 3.19 from Chapter 1, and the general case was described in Exercise 3.11. With the notation of Proposition 1.3, ζ ∈ μ_{m} acts on (x, y) ∈ ℂ^{2} via ζ · (x, y) = (ζx, ζy) and the ring of invariants is

ℂ[x, y]^{μ_{m}} = ℂ[x^{m}, x^{m−1}y, …, xy^{m−1}, y^{m}].

so

U_{σ} ≃ ℂ^{2}/μ_{m} ≃ Spec(ℂ[x^{m}, x^{m−1}y, …, xy^{m−1}, y^{m}]).

On the other hand, from the Hilbert basis of the semigroup S_{σ} = σ^{∨}∩M we obtain the standard toric description

U_{σ} ≃ Spec(ℂ[u, uv, uv^{2}, …, uv^{m}]).

Exercise 1.3 studies the relation between these two representations of the coordinate ring of U_{σ}. ⋄

Example 1.6. Next consider a cone σ with parameters m and k = m − 1, so m = k + 1. (We will express everything in terms of the parameter k in the following.) Unlike the previous example, this is a case we have not encountered previously. Note that k ≡ −1 mod m. Hence by Proposition 1.3, the action of G = N/N′ on ℂ^{2} is given by

ζ · (x, y) = (ζx, ζ^{−1}y).

It is easy to check that the ring of invariants here is

ℂ[x, y]^{μ_{k+1}} = ℂ[x^{k+1}, y^{k+1}, xy].

Moreover we have an isomorphism of rings

φ : ℂ[X, Y, Z]/⟨Z^{k+1} − XY ⟩ ≃ ℂ[x^{k+1}, y^{k+1}, xy]
X ↦ x^{k+1}
Y ↦ y^{k+1}
Z ↦ xy,

so we may identify the toric surface U_{σ} with the variety V(Z^{k+1} − XY) ⊆ ℂ^{3}.

This variety has a singular point at the origin called a rational double point (or Du Val singularity) of type A_{k}. Another standard form of these singularities is given in Exercise 1.4. They are called double points because the lowest degree nonzero term in the defining equation has degree two (that is, the multiplicity of the singularity is two) for all k. The rational double points are the simplest singularities from a certain point of view. The exact definition, which we will give in §3 of this chapter, depends on the notion of a resolution of singularities, which will be introduced shortly. All rational double points appear as singularities of quotient surfaces ℂ^{2}/G where G is a finite subgroup of SU(2, ℂ). There is a complete classification of such points in terms of the Dynkin diagrams of types A_{k}, D_{k}, E_{6}, E_{7}, and E_{8}. The groups corresponding to the diagrams D_{k}, E_{6}, E_{7}, E_{8} are not abelian, however, so by Example 3.21 in Chapter 1, such points do not appear on toric surfaces. We will see one way that the Dynkin diagram A_{k} appears from the geometry of the toric surface U_{σ} in Exercise 1.6, and we will return to this example in §3 of this chapter. More details on these singularities can be found in [30, Chapter VI] and in the article [10]. ⋄ Resolutions of Singularities. A morphism of irreducible varieties F : Y → X induces a mapping F^{∗} : ℂ(X) → ℂ(Y) on function fields. We say F is a birational morphism if F^{∗} is an isomorphism of fields. It is a standard consequence of this condition that there are nonempty Zariski-open subsets U ⊆ Y and V ⊆ X such that F |_{U} gives an isomorphism from U to V. (A proof of a more general statement that implies this is sketched in Exercise 1.5.) However F can fail to be injective on Y \ U. Important examples of birational morphisms are the standard blowups

(1.3)  φ : Bl_{0}(ℂ^{n}) → ℂ^{n}

and the toric blowup morphisms

φ : X_{Σ^{∗}(σ)} → X_{Σ}

generalizing (1.3) constructed in Proposition 3.12 from Chapter 3. In the case of the blowup of the origin in ℂ^{n}, the open sets V = ℂ^{n} \ {0} and U = Bl_{0}(ℂ^{n}) \ φ^{−1}({0}) are isomorphic, while φ^{−1}({0}) ≃ ℙ^{n−1}.

In the basic example φ : Bl_{0}(ℂ^{n}) → ℂ^{n}, ℂ^{n} is itself a smooth variety. However, blowup mappings are often used as follows. Starting from a singular variety X (such as a toric variety X_{Σ} with singular points), we may attempt to find a blowup (or perhaps the composition of a sequence of blowups)

π : Y → X

such that the complexity of the singularities of Y is reduced in some suitable sense. If we can remove or “resolve” the singularities entirely, so the end result Y is smooth, then we have obtained a smooth variety with the same function field as X. Intuitively, we might think of Y as a desingularized, but birationally equivalent, version of the original X. This motivates the following definition.

Definition 1.7. Let X be a singular variety. A resolution of singularities of X is a smooth variety Y and a birational proper morphism π : Y → X.

The existence of resolutions of singularities for curves was established in the 19th century. Resolutions of singularities for surfaces and threefolds over ℂ were established by Zariski in 1939 and 1944, respectively. The existence of a resolution of singularities for all X over ℂ was proved in the landmark 1964 article [19] of Hironaka. We note that one direction of research in modern algebraic geometry seeks to extend results such as this to the case of varieties over general fields, including fields of characteristic p. Unfortunately, Hironaka’s general approach does not work in characteristic p. While existence of resolutions for curves in this case follows by the classical approaches and the result for surfaces and some threefolds was proved by Abhyankar, the existence of resolutions of singularities in characteristic p is still an open problem for varieties of dimension ≥ 4 and for threefolds if p ≤ 5.

One of the interesting and appealing aspects of toric varieties is the way that many questions that are extremely difficult for general varieties admit beautifully

Figure 1. The cones σ_{1}, σ_{2}, τ in Example 1.8

simple and concrete solutions in the toric case. The problem of finding resolutions of singularities is a perfect example. Let us begin to illustrate this by constructing explicit resolutions of the toric surface singularities from Example 1.5 and Example 1.6.

Example 1.8. Consider the rational normal cone of degree m – the affine toric surface U_{σ} for for σ = Cone(e_{2}, me_{1} − e_{2}) studied in Example 1.5. Let Σ′ be the fan consisting of σ and its faces and let Σ be the fan in Figure 1 obtained by inserting a new ray τ = Cone(e_{1}) subdividing σ into two two-dimensional cones:

σ_{1} = Cone(e_{2}, e_{1})
σ_{2} = Cone(e_{1}, me_{1} − e_{2}).

We will now make use of a number of the results from Chapter 3 and all references will be to that chapter. First, the identity mapping on the lattice N is compatible with the fans Σ and Σ′ as in Definition 3.1. By Theorem 3.4, we have a corresponding toric blowup morphism

(1.4)  π : X_{Σ} → X_{Σ′} = U_{σ}.

Note that both σ_{1} and σ_{2} (as well as all of their faces) are smooth cones. Hence Theorem 1.18 implies that X_{Σ} is a smooth surface. In addition, the toric morphism π is proper by Theorem 4.6 since Σ is a refinement of Σ′. Finally, we claim that π is birational. This follows automatically in this case because π restricts to the identity mapping from the torus T_{N} ⊆ X_{Σ} to T_{N} ⊆ X_{Σ′} and these are Zariski-open subsets of the surfaces in question. But of course, we can say more. By the Orbit-Cone Correspondence, it follows that if p_{σ} is the distinguished point corresponding to the two-dimensional cone σ in Σ′ (the singular point of U_{σ} at the origin), then π restricts to an isomorphism

X_{Σ} \ π^{−1}({p_{σ}}) ≃ X_{Σ′} \ {p_{σ}}.

The inverse image E = π^{−1}({p_{σ}}) is the curve on X_{Σ} given by the closure of the T_{N}-orbit O(τ) corresponding to the ray τ. That is, the singular point “blows

Figure 2. The cone σ and the refinement Σ in Example 1.9

up” to E ≃ ℙ^{1} on the smooth surface. It follows that X_{Σ} and the morphism (1.4) give a toric resolution of singularities of the rational normal cone. E is called the exceptional divisor on the smooth surface. We will study more of the geometry of how E sits inside the surface X_{Σ} in §3. ⋄

Example 1.9. We consider the case m = 4 of Example 1.6, for which the surface U_{σ} has a rational double point of type A_{3}. We will leave the verification of our claims, as well as the generalization to all m ≥ 2 to the reader in Exercise 1.6. As in the previous example, write Σ′ for the fan consisting of the cone

σ = Cone(e_{2}, 4e_{1} − 3e_{2})

and its faces. It is easy to find subdivisions of σ yielding collections of smooth cones. The most economical way to do this is to insert three new rays ρ_{1} = Cone(e_{1}), ρ_{2} = Cone(2e_{1} − e_{2}), ρ_{3} = Cone(3e_{1} − 2e_{2}) to obtain a fan Σ consisting of four two-dimensional cones and their faces.

The process is perhaps even clearer if we draw the cones relative to a different basis {f_{1}, f_{2}} for N. Taking f_{1} = e_{2} and f_{2} = e_{1} − e_{2}, the subdivided fan consisting of

σ_{1} = Cone(f_{1}, f_{1} + f_{2})
(1.5)  σ_{2} = Cone(f_{1} + f_{2}, f_{1} + 2f_{2})
σ_{3} = Cone(f_{1} + 2f_{2}, f_{1} + 3f_{2}),
σ_{4} = Cone(f_{1} + 3f_{2}, f_{1} + 4f_{2})

and their faces appears in Figure 2.

You will check that each of these cones is smooth. Hence X_{Σ} is a smooth surface. As in the previous example, since Σ is a refinement of Σ′, we have a toric blowup that yields a resolution of singularities

π : X_{Σ} → X_{Σ′} = U_{σ}.

Also as in the previous example, π restricts to an isomorphism from X_{Σ}\π^{−1}({p_{σ}}) to X_{Σ′} \ {p_{σ}}. In this case, the exceptional divisor E = π^{−1}({p_{σ}}) is the union

E = \overline{O(ρ_{1})} ∪ \overline{O(ρ_{2})} ∪ \overline{O(ρ_{3})}

on X_{Σ}. Each of the curves \overline{O(ρ_{i})} is isomorphic to ℙ^{1}. The first two intersect transversely at the fixed point of the T_{N}-action on X_{Σ} corresponding to the cone σ_{2}, while the second two intersect transversely at the fixed point corresponding to σ_{3}. ⋄

In these examples, we constructed resolutions of affine toric surfaces with just one singular point. The same techniques can be applied to any normal toric surface X_{Σ′}.

Theorem 1.10. Let X_{Σ′} be a normal toric surface. There exists a smooth fan Σ refining Σ′ such that the associated toric morphism π : X_{Σ} → X_{Σ′} is a resolution of singularities.

Proof. If we can show the existence of the smooth fan Σ refining Σ′, the reasoning given in Example 1.8 applies to show that the corresponding toric morphism π is proper and birational, hence a resolution of singularities of X_{Σ′}. Hence, we only need to show that Σ exists.

We will prove this by induction on an integer invariant of fans that measures the complexity of the singularities on the corresponding surfaces. Let σ_{1}, …, σ_{n} denote the two-dimensional cones in a fan Δ. For each i, we will write N_{i} for the sublattice of N generated by the ray generators of σ_{i}. Then we define

s(Δ) = ∑_{i=1}^{n} ([N : N_{i}] − 1) .

If s(Δ) = 0, then n = 0 or [N : N_{i}] = 1 for all i. It is easy to see that this implies that Δ is a smooth fan. Hence X_{Δ} is already smooth and we take this as the base case for our induction.

For the induction step, we assume that the existence of smooth refinements has been established for all fans Δ with s(Δ) < s, and consider a fan Δ′ with s(Δ′) = s. If s ≥ 1, then there exists some nonsmooth cone σ_{i} in Δ′. By Proposition 1.1, there is a basis {e_{1}, e_{2}} for N such that σ_{i} = Cone(e_{2}, me_{1}−ke_{2}) with parameters m > 0, 0 ≤ k < m, and gcd(m, k) = 1. Consider the refinement Δ of Δ′ obtained by subdividing the cone σ_{i} into two new cones

σ′_{i} = Cone(e_{2}, e_{1})
σ′′_{i} = Cone(e_{1}, me_{1} − ke_{2})

with a new one-dimensional cone ρ = Cone(e_{1}). We must show that s(Δ) < s(Δ′) to invoke the induction hypothesis and conclude the proof.

In s(Δ), the terms corresponding to the other cones σ_{j} j ≠ i from Δ′ are unchanged. The cone σ′_{i} is smooth since {e_{1}, e_{2}} is the normalized basis of N relative to σ_{i}. So it contributes a zero term in s(Δ). Finally, we consider the cone σ′′_{i}. We saw in (1.1) that if σ is a cone with parameters m, k then the sublattice N′ generated by the ray generators satisfies [N : N′] = m. Hence we must determine the parameters of the cone σ′′_{i}.

In terms of the basis {e_{1}, e_{2}} for N, the ℤ-linear mapping defined by the matrix

A = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix})

(a “90-degree rotation”) takes σ′′_{i} to Cone(e_{2}, ke_{1} + me_{2}). Since A ∈ GL(2, ℤ), it defines an automorphism of N, and hence σ′′_{i} will have the same parameters as Cone(e_{2}, ke_{1} + me_{2}). But now we apply Lemma 1.2 to put this cone into normal form. There is some integer l, 0 ≤ l < k, such that our cone has parameters k, l. Since k < m, if N′′_{i} is the sublattice generated by the ray generators of σ′′_{i}, then by (1.1),

[N : N′′_{i}] = k < [N : N_{i}] = m.

It follows that

s(Δ) < s(Δ′),

and the proof is complete by induction. □

We will see in the next section that the one-dimensional cones in the refined fan Σ yielding the resolution of X′_{Σ} have a very nice description. Although to this point we have usually only considered the semigroups σ^{∨}∩M defined by the dual cones, we can also consider σ ∩ N for σ themselves. For each two-dimensional cone σ in Σ′, the new cones in Σ are produced by subdividing along the rays through the irreducible elements of the semigroup σ ∩ N. If you look back at Examples 1.8 and 1.9, you will see how this works in two special cases.

A resolution of a non-normal toric surface singularity can be constructed by first saturating the associated semigroup as in Theorem 3.5 from Chapter 1, then applying the results of this section. Similar techniques can also be applied to construct toric blowups giving resolutions of singularities of toric varieties of dimension three and larger. However, we will postpone discussing the higher-dimensional case until Chapter 9.

Exercises for § 1.

1.1. Adapt the usual proof of the integer division algorithm to prove Lemma 1.2.

1.2. In this exercise, you will develop further properties of the parameters m, k in the normal form for cones from Proposition 1.1 and prove part of Proposition 1.4.

154 8. Toric Surfaces

(a) Show that if σ̃ is obtained from a cone σ by parameters m, k by a ℤ-linear mapping of N defined by a matrix in GL(2, ℤ), then the parameter k̃ of σ̃ satisfies either k̃ = k, or kk̃ ≡ 1 mod m. Hint: there is a choice of orientation to be made in the normalization process. Recall that gcd(m, k) = 1 so there are integers m̃, k̃ such that m̃m+k̃k = 1.

(b) Show directly using Proposition 1.3 that the singularities at the origin of the toric varieties U_{σ} and Uσ̃ are isomorphic if σ has parameters m, k and σ̃ has parameters m, k̃ with kk̃ ≡ 1 mod m.

(c) Show that if σ is a cone with parameters m, k, then the dual cone σ^{∨} ⊆ M_{ℝ} has parameters m, m − k. Hint: Use the normal form for σ, write down σ^{∨} in the corresponding dual coordinates in M, then change coordinates in M to normalize σ^{∨}.

1.3. With the notation in Example 1.5, show that

ℂ[u, uv, uv^{2}, …, uv^{m}] ≃ ℂ[x^{m}, x^{m−1}y, …, xy^{m−1}, y^{m}]

under u ↦ x^{m} and v ↦ y/x, and use Proposition 1.3 to explain where these identifications come from in terms of the semigroup S_{σ}. Hint: We have u = χ^{e_{1}^{∗}} and v = χ^{e_{2}^{∗}} where {e_{1}, e_{2}} is the normalized basis for N and {e_{1}^{∗}, e_{2}^{∗}} is the dual basis for M.

1.4. In Example 1.6, we gave one form of the rational double point of type A_{k}, namely the singular point at (0, 0, 0) on the surface V = V(Z^{k+1} − XY) ⊆ ℂ^{3}. Another commonly used normal form for this type of singularity is the singular point at (0, 0, 0) on the surface W = V(X^{k+1} + Y^{2} + Z^{2}). Show that V and W are isomorphic as affine varieties, hence the singularities at the origin are analytically equivalent. Hint: there is a linear change of coordinates in ℂ^{3} that shows this.

1.5. Let X, Y be irreducible varieties. Consider the class of morphisms F : U → Y where U is a nonempty Zariski-open subset of X. We will say F : U → Y and F′ : V → Y are equivalent if F |_{U∩V} = F′|_{U∩V}. A rational map F : X – – → Y is by definition an equivalence class of such morphisms under this notion of equivalence. A rational mapping F : X – – → Y is birational if it has a rational inverse G : Y – – → X (that is F ∘G = id and G ∘ F = id where the compositions are defined).

(a) Show that a birational mapping F induces an isomorphism of function fields F^{∗} : ℂ(Y) ≃ ℂ(X).

(b) Show that if F is birational, then there are nonempty Zariski-open subsets U ⊆ X and V ⊆ Y such that F restricts to an isomorphism from U to V. (This applies in particular when F is a birational morphism, hence a mapping with domain equal to X.)

1.6. In this exercise, you will check several of the claims made in Example 1.9 and show how to extend the results there to the case σ = Cone(e_{2}, me_{1} − (m − 1)e_{2}) for general m.

(a) Check that each of the four cones in (1.5) is smooth, so that the toric surface X_{Σ} is smooth by Theorem 1.18 from Chapter 3.

(b) For general m, show how to insert new rays ρ_{i} to subdivide σ and obtain a fan Σ whose associated toric surface is smooth. Try to do this with as few new rays as possible. Hence we obtain toric resolutions of singularities π : X_{Σ} → U_{σ} for all m.

(c) Identify the inverse image C = π^{−1}({p_{σ}}) in general. For instance, how many irreducible components does C have? How are they connected? Hint: One way to represent the structure is to draw a graph with vertices corresponding to the components and to connect two vertices by an edge if and only if the components intersect on X_{Σ}. Do you notice a relation between this graph and the Dynkin diagram A_{k} = A_{m−1} mentioned before? We will discuss the relation in detail in §3 of this chapter.

2. Hirzebruch-Jung Continued Fractions

Let σ ⊆ N_{ℝ} be a cone with parameters m, k as in Proposition 1.1. The process of constructing the resolution of singularities for U_{σ} as in Theorem 1.10 has an interesting interpretation in terms of a modification of the continued fraction expansion of a rational number from elementary number theory. We will define a variant known as the Hirzebruch-Jung continued fraction in this section, and show how the operations from §1 can be interpreted in terms of these continued fractions. We will also use them to draw some additional conclusions about the sequences of toric blowups involved in resolutions of toric surface singularities.

Ordinary Continued Fractions. If m > k > 0 are integers, then the ordinary continued fraction expansion of m/k may be obtained by performing the same sequence of integer divisions employed in the Euclidean algorithm for the gcd. Starting with k_{−1} = m and k_{0} = k, we write a_{i} for the quotient and k_{i} for the remainder at each step, so that the ith division is given by

(2.1)  k_{i−2} = a_{i}k_{i−1} + k_{i},

where 0 ≤ k_{i} < k_{i−1}.

Say k_{n−1} is the final nonzero remainder (which equals gcd(m, k)). The resulting equations can be rearranged:

m/k = a_{1} + k_{1}/k
k/k_{1} = a_{2} + k_{2}/k_{1}

⋮ (2.2)

k_{n−3}/k_{n−2} = a_{n−1} + k_{n−1}/k_{n−2}
k_{n−2}/k_{n−1} = a_{n},

and spliced together to form the continued fraction

m/k = a_{1} + \frac{1}{a_{2} + _{\frac{1}{··· + \frac{1}{an}}}} .

For obvious typographical reasons, it is desirable to have a more compact way to represent these expressions. We will use the notation

(2.3)  m/k = [a_{1}, a_{2}, …, a_{n}]

for the ordinary continued fraction. In the context of continued fraction expansions, the a_{i} are called the partial quotients of m/k; the truncated continued fractions

c_{i} = [a_{1}, a_{2}, …, a_{i}], 1 ≤ i ≤ n

are called the convergents of m/k. For future reference, we mention that there are recurrences for the numerators and denominators of the convergents.

Proposition 2.1. Let m > k > 0 be integers. Define sequences p_{i} and q_{i} recursively as follows. First set

p_{0} = 1  q_{0} = 0
p_{1} = a_{1}  q_{1} = 1.

Then, for all 2 ≤ i ≤ n, let

(2.4)  p_{i} = p_{i−2} + a_{i}p_{i−1}
q_{i} = q_{i−2} + a_{i}q_{i−1}.

The convergents of the ordinary continued fraction expansion for m/k are given by

c_{i} = p_{i}/q_{i}, 1 ≤ i ≤ n.

Moreover, gcd(p_{i}, q_{i}) = 1 for all i.

The proof is left to the reader as Exercise 2.1. Ordinary continued fraction expansions can also be constructed for arbitary real numbers α by modifying (2.2) appropriately. If α is irrational, the expansion will be infinite (see Exercise 2.2).

Hirzebruch-Jung Continued Fractions. What do continued fractions have to do with toric surfaces? To answer this, let us consider what happens when we start with the singular point at the origin on the affine toric surface U_{σ} for a cone σ in normal form with parameters m, k, and construct a resolution of singularities using a process following the proof of Theorem 1.10. The first step in the resolution is to refine the cone σ to a fan Σ containing the two-dimensional cones

σ′ = Cone(e_{2}, e_{1}) and σ′′ = Cone(e_{1}, me_{1} − ke_{2}).

The first is smooth, but the second may not be. However, we saw that the cone σ′′ has parameters k, k_{1} where

m = b_{1}k − k_{1},

b_{1} > 0, 0 ≤ k_{1} < k (as in Lemma 1.2). We used slightly different notation before, writing s rather than b_{1}. The new notation will help us keep track of what happens if we now continue the process and refine the cone σ′′.

Just as before, using the normalized basis for N relative to σ′′ we insert a new ray and obtain a new smooth cone and a second, possibly nonsmooth cone with parameters k_{1}, k_{2}, where

k = b_{2}k_{1} − k_{2}

using Lemma 1.2.

This lemma is a sort of modified division algorithm, and applying it repeatedly yields a modified Euclidean algorithm

m = b_{1}k − k_{1}
k = b_{2}k_{1} − k_{2}

⋮ (2.5)

k_{r−3} = b_{r−1}k_{r−2} − k_{r−1}
k_{r−2} = b_{r}k_{r−1}

that computes the parameters of the new cones produced as we successively subdivide to produce the fan giving the resolution of singularities. The process terminates with k_{r} = 0 for some r as shown, since as in the usual Euclidean algorithm, the k_{i} are a strictly decreasing sequence of nonnegative numbers. Since 0 ≤ k_{i+1} < k_{i} for all i by Lemma 1.2, we will have b_{i} ≥ 2 for all i.

Now, rearranging these equations just as we did before in (2.2) gives a new type of continued fraction expansion for the rational number m/k, with minus signs replacing the plus signs:

(2.6)  m/k = b_{1} − \frac{1}{b_{2} − _{\frac{1}{··· − \frac{1}{br}}}} .

This is called the Hirzebruch-Jung continued fraction expansion. We will abbreviate it as

m/k = [[b_{1}, b_{2}, …, b_{r}]].

The integers b_{i} are called the partial quotients of the Hirzebruch-Jung continued fraction. The truncated Hirzebruch-Jung continued fractions:

d_{i} = [[b_{1}, b_{2}, …, b_{i}]], 1 ≤ i ≤ r

are the convergents. We have d_{r} = [[b_{1}, b_{2}, …, b_{r}]] = m/k.

Example 2.2. Consider the rational number 17/11. The ordinary continued fraction expansion is

17/11 = [1, 1, 1, 5].

On the other hand, the Hirzebruch-Jung continued fraction expansion is

17/11 = [[2, 3, 2, 2, 2, 2]].

Note that the partial quotients and the lengths are both different. As noted above, all b_{i} ≥ 2 in a Hirzebruch-Jung continued fraction. Each of these expansions determines the other and there are methods for computing the Hirzebruch-Jung partial quotients b_{j} in terms of the ordinary partial quotients a_{i}. See, for instance, [20, p. 257] and Exercise 2.3 below. ⋄ Hirzebruch-Jung Continued Fractions and Resolutions. The process of computing the Hirzebruch-Jung continued fraction of m/k by the modified Euclidean algorithm (2.5) also yields a very convenient computational method for producing the fan Σ such that π : X_{Σ} → U_{σ} is a resolution of singularities. The following process for computing the numerator and denominators of the d_{i} is the direct generalization of Proposition 2.1.

Proposition 2.3. Let m > k > 0 be integers. Define sequences p_{i} and q_{i} recursively as follows. First set

p_{0} = 1  q_{0} = 0
p_{1} = b_{1}  q_{1} = 1.

Then, for all 2 ≤ i ≤ r, let

(2.7)  p_{i} = −p_{i−2} + b_{i}p_{i−1}
q_{i} = −q_{i−2} + b_{i}q_{i−1}.

The convergents of the Hirzebruch-Jung continued fraction expansion for m/k are given by

d_{i} = p_{i}/q_{i}, 1 ≤ i ≤ r.

There are two differences between the recurrences in (2.7) and those in (2.4). First, note the minus signs on the right of each equation. Second, the b_{i} from the Hirzebruch-Jung continued fraction are used in place of the a_{i} from the ordinary continued fraction. You will adapt the proof of Proposition 2.1 to prove this in Exercise 2.4. We next show how to compute the fan Σ refining the cone σ using this information.

Theorem 2.4. Let σ be a cone with parameters m, k and let {e_{1}, e_{2}} be the normalized basis for N relative to σ. Let v_{0} = e_{2}. Using the integers p_{i} and q_{i} from Proposition 2.3, construct vectors

v_{i} = p_{i−1}e_{1} − q_{i−1}e_{2}, 1 ≤ i ≤ r + 1,

and cones

σ_{i} = Cone(v_{i−1}, v_{i}), 1 ≤ i ≤ r + 1.

Then

(a) Each σ_{i} is a smooth cone and v_{i−1}, v_{i} are its ray generators.

(b) For each i, σ_{i+1} ∩ σ_{i} = Cone(v_{i}).

(c) We have σ_{1} ∪ ⋯ ∪ σ_{r+1} = σ so the fan Σ consisting of the σ_{i} and their faces gives a refinement of σ.

(d) The toric morphism π : X_{Σ} → U_{σ} is a resolution of singularities.

Proof. Both statements in part a will follow from the equations

(2.8)  p_{i−1}q_{i} − p_{i}q_{i−1} = 1, 1 ≤ i ≤ r.

We will prove this by induction on i. The base case i = 1 follows directly from the initialization of the p_{i}, q_{i} sequences in Proposition 2.3. Now assume that the result has been proved for i ≤ s, and consider i = s + 1. Using the recurrences (2.7) we have

p_{s}q_{s+1} − p_{s+1}q_{s} = p_{s}(−q_{s−1} + b_{s}q_{s}) − (−p_{s−1} + b_{s}p_{s})q_{s}
= p_{s−1}q_{s} − p_{s}q_{s−1}
= 1

by the induction hypothesis. The fact that v_{i} are primitive follows from (2.8), since these equations show gcd(p_{i}, q_{i}) = 1 for all i.

For part b, we note that the ratio −q_{i−1}/p_{i−1} represents the slope of the line through v_{i} in the coordinate system relative to the normalized basis {e_{1}, e_{2}} for σ. We claim that these slopes form a strictly decreasing sequence for i ≥ 0, which will imply the statement in part b.

It is an easy consequence of the recurrences (2.7) and the fact that b_{i} ≥ 2 for all i that the p_{i} and q_{i} are increasing sequences of nonnegative numbers. You will show this in Exercise 2.5.

This applies to the slopes as follows. From (2.8),

−q_{i}/p_{i} = −q_{i−1}/p_{i−1} − 1/(p_{i−1}p_{i}) < −q_{i−1}/p_{i−1},

since p_{i−1}p_{i} > 0. Thus, σ_{i} and σ_{i+1} lie on opposite sides of Cone(v_{i}) for each i, and part b follows.

Part c follows from part b noting that v_{0} = e_{2} and p_{r}/q_{r} = m/k, so v_{r+1} = me_{1} − ke_{2}. Hence the cones σ_{i} fill out σ.

Part d follows from part c by the same reasoning used in the examples in §1 of this chapter. □

Example 2.5. Consider the cone σ = Cone(e_{2}, 7e_{1} − 5e_{2}) in normal form. To construct the resolution of the singularity of the affine toric surface U_{σ}, we simply compute the Hirzebruch-Jung continued fraction expansion of the rational number m/k = 7/5 using the modified Euclidean algorithm:

7 = 2 · 5 − 3
5 = 2 · 3 − 1
3 = 3 · 1.

Hence b_{0} = b_{1} = 2, b_{2} = 3, and

(2.9)  7/5 = [[2, 2, 3]].

Then from Proposition 2.3 we have

p_{0} = 1  q_{0} = 0
p_{1} = 2  q_{1} = 1
p_{2} = −p_{0} + b_{2}p_{1} = 3  q_{2} = −q_{0} + b_{2}q_{1} = 2
p_{3} = −p_{1} + b_{3}p_{2} = 7  q_{3} = −q_{1} + b_{3}q_{2} = 5.

By Theorem 2.4, the cones in the refinement Σ of σ giving the resolution of singularities are

σ_{1} = Cone(e_{2}, e_{1})
(2.10)  σ_{2} = Cone(e_{1}, 2e_{1} − e_{2})
σ_{3} = Cone(2e_{1} − e_{2}, 3e_{1} − 2e_{2})
σ_{4} = Cone(3e_{1} − 2e_{2}, 7e_{1} − 5e_{2}).

We can also see an interesting connection between this Hirzebruch-Jung continued fraction and the corresponding continued fraction expansion for 7/3, where 5 · 3 ≡ 1 mod 7 as in Proposition 1.4. We leave it to the reader to check that

7/3 = [[3, 2, 2]],

in which the partial quotients are the same as those in (2.9), but listed in reverse order. We will return to this point shortly. ⋄

You will see several other explicit examples of this process in Exercises 2.8 and 2.9. Our next result gives an alternative, more intrinsic, way to understand the vectors v_{i} in Theorem 2.4. Note that as is true for σ^{∨} ∩ M, the set σ ∩ N has the structure of an additive semigroup. We define its irreducible elements and the Hilbert basis as in Proposition 2.20 from Chapter 1. See Figure 2 in Example 1.9 for an example.

Theorem 2.6. Let σ = Cone(e_{2}, me_{1} − ke_{2}) be in normal form. Then

S = {v_{0}, v_{1}, …, v_{r+1}}

constructed in Theorem 2.4 is precisely the Hilbert basis (the set of irreducible elements) in the semigroup σ ∩ N.

Proof. Since each cone σ_{i} is smooth, σ_{i} ∩ N is generated by its ray generators {v_{i−1}, v_{i}}. Since σ = σ_{1} ∪ ⋯ ∪ σ_{r+1}, it follows that S generates σ ∩ N.

We claim next that all the v_{i} are irreducible elements of σ ∩ N. This is clear for v_{0} = e_{2} and v_{r+1} = me_{1} − ke_{2} since they are the ray generators for σ. If 1 ≤ i ≤ r and v_{i} is not irreducible, then v_{i} would have to be a linear combination of the vectors in S\{v_{i}} with nonnegative integer coefficients. We will now use the result of Exercise 2.5, namely that the sequences p_{i} and q_{i} are increasing sequences of nonnegative numbers. This means that if

v_{i} = p_{i}e_{1} − q_{i}e_{2} = ∑_{j≠i} c_{j}v_{j} = (∑_{j≠i} c_{j}p_{j}) e_{1} − (∑_{j≠i} c_{j}p_{j}) e_{2}

with c_{j} ≥ 0 in ℤ, then

p_{i} = ∑_{j≠i} c_{j}p_{j}, q_{i} = ∑_{j≠i} c_{j}q_{j}.

But the fact that p_{i} and q_{i} are increasing and c_{j} ≥ 0 implies that c_{j} = 0 for all j > i. But this would imply that v_{i} is a linear combination with nonnegative integer coefficients of the vectors in {v_{0}, …, v_{i−1}}. This contradicts the fact from the proof of Theorem 2.4 that the slopes of the v_{i} are strictly decreasing, and it follows that all the v_{i} are irreducible elements of the semigroup σ ∩ N.

Finally, we must show that there are no other irreducible elements in σ ∩ N. But this follows from what we have already said. Since σ = σ_{1} ∪ ⋯ ∪ σ_{r+1}, if v is any irreducible element, then v ∈ σ_{i} ∩ N for some i. But then v = c_{i−1}v_{i−1} + c_{i}v_{i} for some c_{i−1}, c_{i} ≥ 0 in ℤ. Thus v is irreducible only if c_{i−1} = 1 and c_{i} = 0 or vice versa. □

Next, we will show how the irreducible elements v_{i} in σ ∩ N determine the partial quotients in the Hirzebruch-Jung continued fraction expansion of m/k.

Theorem 2.7. Let σ = Cone(me_{1} − ke_{2}) be in normal form, and let

m/k = [[b_{1}, b_{2}, …, b_{r}]].

The vectors {v_{0}, v_{1}, …, v_{r+1}} constructed in Theorem 2.4 satisfy

(2.11)  v_{i−1} + v_{i+1} = b_{i}v_{i}

for all i, 1 ≤ i ≤ r.

Proof. By the recurrences (2.7),

v_{i−1} + v_{i+1} = (p_{i−2}e_{1} − q_{i−2}e_{2}) + (p_{i}e_{1} − q_{i}e_{2})
= (p_{i−2} + p_{i})e_{1} − (q_{i−2} + q_{i})e_{2}
= b_{i}(p_{i−1}e_{1} − q_{i−1}e_{2})
= b_{i}v_{i},

which is what we wanted to show. □

In the next section we will see several important consequences of the equations (2.11) connected with the geometry of the smooth toric surface X_{Σ}. To conclude this section, we will show that the pattern noted at the end of Example 2.5 is a general phenomenon about Hirzebruch-Jung continued fractions, and this can be proved in a nice way from the properties of the associated toric surfaces we have developed.

Proposition 2.8. Let 0 < k, k̃ < m and assume kk̃ ≡ 1 mod m. If the Hirzebruch-Jung continued fraction expansion of m/k is

m/k = [[b_{1}, b_{2}, …, b_{r}]],

then the Hirzebruch-Jung continued fraction expansion of m/ k̃ is

m/ k̃ = [[b_{r}, b_{r−1}, …, b_{1}]].

Proof. Let σ = Cone(e_{2}, me_{1} − ke_{2}) and σ̃ = Cone(e_{2}, me_{1} − k̃e_{2}) be the corresponding cones in normal form. Since kk̃ ≡ 1 mod m, there is an integer m̃ such that mm̃ + kk̃ = 1. The ℤ-linear mapping φ : N → N defined with respect to the basis {e_{1}, e_{2}} by the matrix

A = (\begin{matrix}k̃ & m \\ m̃ & −k \end{matrix})

is bijective, maps σ̃ to σ, and is orientation-reversing so φ(me_{1} − k̃e_{2}) = e_{2} and φ(e_{2}) = me_{1} − ke_{2}. If we apply Theorem 2.4 to σ, then we obtain vectors v_{i} satisfying the equations

v_{i−1} + v_{i+1} = b_{i}v_{i}

for all 1 ≤ i ≤ r. We claim that when we apply the mapping φ^{−1} defined by the inverse of the matrix A above, then the vectors v_{i} are taken to corresponding vectors ṽ_{i} for the cone σ̃. But since φ and φ^{−1} are orientation-reversing, the partial quotients in the Hirzebruch-Jung continued fraction will be listed in the opposite order. You will complete the proof of this assertion in Exercise 2.7. □

Exercises for § 2.

2.1. In this problem we will consider the statement in Proposition 2.1.

(a) Show that the sequences p_{i} and q_{i} satisfy

p_{i−1}q_{i} − p_{i}q_{i−1} = −1

for all i. Hint: see the proof of part a of Theorem 2.4.

(b) Prove the proposition. Hint: Argue by induction on the length n ≥ 1 of the expansion. The formal expression [a_{1}, a_{2}, …, a_{i}] is well-defined even if the a_{l} are not integers, so we can write

[a_{1}, a_{2}, …, a_{i−1}, a_{i}] = [a_{1}, a_{2}, …, a_{i−1} + 1/a_{i}].

Then use the recurrences in (2.4). For the statement about the gcd(p_{i}, q_{i}), use part a.

(c) Suppose we modify the initialization and the recurrences (2.4) as follows. Let

r_{−1} = 1  s_{−1} = 0
r_{0} = 0  s_{0} = 1.

2. Hirzebruch-Jung Continued Fractions 163

Then, for all 1 ≤ i ≤ n, compute

r_{i} = r_{i−2} − a_{i}r_{i−1}
s_{i} = s_{i−2} − a_{i}s_{i−1}

(note the change in sign!) What is true about r_{i}m + s_{i}k for all i? What do we get with i = n? Hint: This fact is the basis for the Extended Euclidean Algorithm.

2.2. In this exercise, we will study the analogs of the continued fraction expansions from (2.3) for arbitrary reals. If α is a real number, then we say

α = [a_{1}, a_{2}, a_{3}, …]

if, by analogy with (2.2), α = a_{1} + α_{1}, where a_{1} is the greatest integer less than or equal to α and 0 ≤ α_{1} < 1. Then for all i ≥ 1,

1/α_{i} = a_{i+1} + α_{i+1},

where at each step a_{i} > 0 is an integer, and 0 ≤ α_{i} < 1 is the fractional part of 1/α_{i−1}. Most computer algebra systems can compute continued fraction expansions.

(a) Show that the result of Proposition 2.1 holds even when the continued fraction expansion does not terminate.

(b) Compute the continued fraction expansions of α = \sqrt{2} and α = \sqrt{3}. Do you notice something interesting?

(c) Show that if the continued fraction expansion of α is eventually periodic, then α satisfies an irreducible quadratic polynomial with rational coefficients. The converse of this statement is also true, but somewhat more difficult to prove.

(d) Compute a_{0}, …, a_{20} for α = e. Do you notice something interesting?

2.3. A key ingredient for a method for converting ordinary continued fraction expansions to Hirzebruch-Jung continued fractions is the following observation.

(a) Given a_{0}, a_{1} > 0 and a variable x, show that as rational expressions

[a_{1}, a_{2}, x] = [[a_{1} + 1, (2)^{a_{2}−1}, x + 1]],

where for any l ≥ 0, (2)^{l} denotes a string of l 2’s. Hint: argue by induction on a_{1}.

(b) Show how this observation yields the equality

[1, 1, 1, 5] = [[2, 3, 2, 2, 2, 2]]

from Example 2.2.

2.4. Prove Proposition 2.3 by adapting the proof from Exercise 2.1.

2.5. Show that the sequences p_{i} and q_{i} from (2.7) are increasing sequences of nonnegative numbers. Hint: recall that b_{i} ≥ 2 for all i.

2.6. Verify that equations (2.11) hold for the vectors v_{i} computed in Example 2.5. (See (2.10).)

2.7. Verify the last claim in the proof of Proposition 2.8.

2.8. In §1, we constructed several resolutions of singularities in a rather ad hoc way. In this exercise, we will see that the resolutions given by Theorem 2.4 are the same as what we saw before.

(a) Show that the Hirzebruch-Jung continued fraction method gives exactly the same resolution of the singular point p_{σ} for σ with parameters m, 1 as that given in Example 1.8.

(b) Do the same for the resolutions of the singular points for σ with parameters m, k = m − 1 (the rational double points from Example 1.9). Hint: You will want to start by showing that

m/(m − 1) = [[2, 2, …, 2]],

where there are m − 1 2’s.

2.9. In this exercise, you will apply the results of this section to the weighted projective planes ℙ(q_{0}, q_{1}, q_{2}) from Chapter 2 and Example 1.16 in Chapter 3.

(a) Construct a resolution of singularities for any ℙ(1, 1, q_{2}), where q_{2} ≥ 2. What smooth toric surface is obtained in this way? A complete classification of the smooth complete toric surfaces will be developed in §3 of this chapter.

(b) Do the same for ℙ(1, q_{1}, q_{2}) in general.

3. Smooth Toric Surfaces

Classification of Smooth Toric Surfaces. We will begin this section by using the results of §§1 and 2 to give a complete classification of the smooth complete toric surfaces. Our theorem will say that these are all obtained by toric blowups from either ℙ^{2}, ℙ^{1} × ℙ^{1}, or one of the Hirzebruch surfaces H_{r} with r ≥ 2 from Example 1.15 in Chapter 3. The proof will be based on the following facts.

First, it follows from Proposition 3.12 of Chapter 3 that if σ = Cone(v_{1}, v_{2}) is a smooth cone and we refine σ by inserting the new one-dimensional cone τ = Cone(v_{1} + v_{2}), then on the resulting toric surface, the smooth point p_{σ} will be blown up to a copy of ℙ^{1}.

For the second ingredient of the proof, we introduce the following notation. If Σ is a smooth complete fan, then list the ray generators of the two-dimensional cones in Σ as v_{0}, v_{1}, …, v_{d−1} in clockwise order around the origin in N_{ℝ}, and we will consider the indices as integers modulo d, so v_{d} = v_{0}. Then we have the following statement parallel to (2.11).

Lemma 3.1. Let v_{0}, …, v_{d} be the ray generators for a smooth complete fan Σ in N_{ℝ} ≃ ℝ^{2}. For all i, there exist integers b_{i}, i = 0, …, d − 1 such that

(3.1)  v_{i−1} + v_{i+1} = b_{i}v_{i}.

Proof. All the cones in Σ are assumed to be smooth. Hence {v_{i−1}, v_{i}} is one ℤ-basis for N and we can write

v_{i+1} = a_{i}v_{i−1} + b_{i}v_{i}

for some integers a_{i}, b_{i}. By our choice of indices {v_{i}, v_{i+1}} is another ordered ℤ-basis for N with the same orientation as {v_{i−1}, v_{i}}. Hence

det (\begin{matrix}0 & a_{i} \\ 1 & b_{i} \end{matrix}) = +1.

It follows that a_{i} = −1 and v_{i−1} + v_{i} = b_{i}v_{i} as claimed. □

A consequence of the proof of this fact and the results of §2 is the following statement.

Lemma 3.2. Let Σ be a smooth fan which is a refinement of a smooth cone σ. Then Σ is obtained from σ by a sequence of star subdivisions as in Definition 3.11 from Chapter 3.

Proof. Say there are r two-dimensional cones in Σ and the ray generators in are v_{0}, v_{1}, …, v_{r}, listed clockwise starting from v_{0}. We argue by induction on r. If r = 1, there is only one cone in Σ and there is nothing to prove. Assume the result has been proved for all Σ with r = s two-dimensional cones, and consider the case r = s + 1. The proof of Lemma 3.1 works locally in the fan, so under the hypotheses here, for all i, 1 ≤ i ≤ s, there exist integers b_{i} such that (3.1) holds. Note that σ is strongly convex so b_{i} > 0 for all i. We claim that there exists some i such that b_{i} = 1. If not, that is, if b_{i} ≥ 2 for all i, then as in §2, the Hirzebruch-Jung continued fraction

[[b_{1}, b_{2}, …, b_{s}]]

represents a rational number m/k and the cone σ has parameters m, k with m > k > 0. But then m ≥ 2, which would contradict the assumption that σ is a smooth cone.

Hence there exists an i, 1 ≤ i ≤ s, such that

v_{i−1} + v_{i+1} = v_{i}.

In this situation, Cone(v_{i−1}, v_{i+1}) is also smooth. You will verify this claim in Exercise 3.1. Moreover, Cone(v_{i−1}, v_{i}) and Cone(v_{i}, v_{i+1}) are precisely the cones in the star subdivision of Cone(v_{i−1}, v_{i+1}). Hence the proof is complete by induction. □

We are now ready to state our classification theorem.

Theorem 3.3. Every smooth complete toric surface X_{Σ} is obtained from either

ℙ^{2}, ℙ^{1} × ℙ^{1}, or H_{r}, r ≥ 2

by a finite sequence of blowups at fixed points of the torus action.

Proof. We label the ray generators of the fan Σ as in Lemma 3.1. As in the proof of Lemma 3.2, if b_{i} = 1 in (3.1)for some i, then our surface is a blowup of the surface corresponding to the fan where v_{i} is removed. Hence we only need to consider the case in (3.1) where b_{i} ≠ 1 for all i.

Suppose first that v_{j} = −v_{i} for some i < j. We relabel the vertices to make v_{j} = −v_{1} for some j. Note that j > 2 since the cones must be strongly convex. Then from (3.1),

v_{0} = −v_{2} + b_{1}v_{1}.

Figure 3. The ray generators v_{0}, v_{1}, v_{2}, v_{j} when v_{j} = −v_{1}

These ray generators are shown in Figure 3, relative to the ℤ-basis {v_{1}, v_{2}} for N. Comparing this with the fans for H_{r} and ℙ^{1} × ℙ^{1} (Figures 3 and 4 from Chapter 3, §1), we see that Σ must be a refinement of the fan for H_{r} if r = b_{1} > 2. The same follows if b_{1} < −2 and r = |b_{1}| (see Exercise 3.2). If b_{1} = 0, then we have a refinement of the fan for ℙ^{1} × ℙ^{1}. Hence the theorem follows from Lemma 3.2 in the case v_{j} = −v_{1} for some j.

Now suppose v_{j} ≠ −v_{i} for all j > i. Consider the ℤ-basis {v_{1}, v_{2}} for N and the possible locations of v_{3}. We will use the usual terminology for the quadrants in N_{ℝ} relative to the axes of the real coordinate system with the real span of v_{1} playing the role of the y-axis and the real span of v_{2} playing the role of the x-axis. However, the ordered basis {v_{1}, v_{2}} has the opposite of the usual orientation. This means, for instance, that the open fourth quadrant will consist of vectors

w = av_{1} + bv_{2}

with a < 0 and b > 0. Since Cone(v_{2}, v_{3}) must be strongly convex, v_{3} must lie in the interior of either the third or fourth quadrant of the plane If v_{3} is in the third quadrant, then from (3.1),

v_{3} = −v_{1} + b_{2}v_{2},

where b_{2} < 0.

If b_{2} = −1, then Cone(v_{3}, v_{1}) is smooth. It follows from our simplifying assumptions based on Lemma 3.2 that v_{0} = v_{3} and we see from Figure 2 of Chapter 1 that Σ must be the standard fan for ℙ^{2}. (If we did not invoke Lemma 3.2, the conclusion would be that Σ was a refinement of the fan for ℙ^{2}.) The theorem follows in this case.

We claim that the case b_{2} < −1 cannot occur under the assumption v_{j} ≠ −v_{j}. We may use the ℤ-basis f_{1} = −v_{2}, f_{2} = v_{1} for N. Writing m = −b_{2}, we have

Cone(v_{3}, v_{1}) = Cone(mf_{1} − f_{2}, f_{2}),

which is in normal form with parameters m, 1, but with the reverse of the usual orientation. The other vectors v_{4}, …, v_{d} = v_{0} come from a smooth refinement of this cone. Hence by Proposition 2.8, we must have

m = m/1 = [[b_{d}, b_{d−1}, …, b_{4}]].

However, by the modified division algorithm from §2, this implies that d = 4 and b_{4} = −b_{2} = m. We have

−b_{2}v_{0} = b_{4}v_{0} = b_{4}v_{4} = v_{3} + v_{5} = v_{3} + v_{1} = b_{2}v_{2}.

This would imply v_{2} = −v_{0}, which is a contradiction.

Finally, if v_{3} is in the fourth quadrant, consider the last ray generator v_{j}, j ≥ 3 in the fourth quadrant, counting clockwise from v_{2}. We have

(3.2)  v_{j} = av_{2} − bv_{1},

where a, b > 0. The next vector v_{j+1} must lie to the left of the line containing v_{1} (since we are assuming v_{j+1} ≠ −v_{1}). Hence

(3.3)  v_{j+1} = −cv_{2} + dv_{1},

where c > 0. The case d = 0 cannot occur under our assumption v_{j} ≠ −v_{i}. We claim that d ≠ 0 also leads to a contradiction. Note that by the hypotheses Cone(v_{j}, v_{j+1}) is smooth and has the same orientation as Cone(v_{1}, v_{2}). Hence the constants a, b, c, d from (3.2) and (3.3) satisfy

det ( \begin{matrix}−b & d \\ a & −c \end{matrix} ) = bc − ad = 1.

But a, b, c, d are all integers and a, b, c > 0. If d < 0 then the determinant is strictly larger than 1. If d > 0, then v_{j+1} lies in the second quadrant and

v_{i} + v_{j+1} = (a − c)v_{2} + (d − b)v_{1},

lies in the third quadrant, so a − c < 0 and d − b < 0. But then c > a > 0 and b > d > 0. Hence bc − ad must be strictly larger than 1 again. You will check the details of the last part of this argument in Exercise 3.3. □

Since there is also a Hirzebruch surface H_{1}, the statement of this theorem might seem puzzling. The reason that H_{1} is not included is that this surface is actually a blowup of ℙ^{2}. See Exercise 3.4.

The problem of classifying all higher-dimensional toric varieties is much more difficult. No complete classifications are known, although results for fans with small numbers of one-dimensional cones relative to the dimension of N_{ℝ} have been developed in [3].

Divisors on a Smooth Toric Surface. Our next results will deal with the geometry of divisors on smooth complete toric surfaces. We begin by summarizing how the general facts on divisors and line bundles on toric varieties from Chapters 5 and 6 specialize to the case of a smooth complete toric surface X_{Σ}. (**add references later**)

• First, since X_{Σ} is smooth, Weil and Cartier divisors coincide.

• The group Pic(X_{Σ}) of line bundles modulo isomorphism coincides with the group of divisor classes Cl(X_{Σ}), and there is an exact sequence

(3.4)  0 ⟶ M ⟶ ⊕_{i=1}^{d}ℤ · D_{i} ⟶ Pic(X_{Σ}) ⟶ 0,

where the D_{i} are the divisors corresponding to the one-dimensional cones Cone(v_{i}) in Σ and d is the number of edges. The mapping to ⊕_{i=1}^{d}ℤ · D_{i} takes u ∈ M to

div(χ^{u}) = ∑_{i=1}^{d}⟨u, v_{i}⟩D_{i}.

The mapping to Pic(X_{Σ}) is the map sending D to the isomorphism class of the line bundle O(D).

• From this exact sequence we have an isomorphism

Pic(X_{Σ}) ≃ ℤ^{d−2}.

• The canonical divisor class on X_{Σ} is the class of

K = −(D_{1} + ⋯ + D_{d}),

where as before the D_{i} are the divisors corresponding to the one-dimensional cones in Σ.

Example 3.4. For the Hirzebruch surface H_{r}, use the standard fan as in Figure 4 from Chapter 3, and let D_{1} correspond to Cone(e_{2}), D_{2} correspond to Cone(e_{1}), D_{3} correspond to Cone(−e_{2}) and D_{4} correspond to Cone(−e_{1} + re_{2}). Then

Pic(H_{r}) ≃ ⊕_{i=1}^{4}ℤ · D_{i}/(ℤ · (D_{2} − D_{4}) ⊕ ℤ · (D_{1} − D_{3} + rD_{4})).

Hence D_{4} ≡ D_{2} and D_{3} ≡ D_{1} + rD_{2}, so it follows that Pic(H_{r}) ≃ ℤ^{2} is freely generated by the classes of D_{1} and D_{2}. Moreover, the canonical divisor is

K = −(D_{1} + D_{2} + D_{3} + D_{4}) ≡ −2D_{1} − (r + 2)D_{2}.
⋄

The Intersection Product. A fundamental feature of the theory of smooth algebraic surfaces is the intersection product on divisors or divisor classes. Let C and D be two curves on a smooth surface S having no irreducible components in common, so C ∩ D is a finite collection of points. At each p ∈ C ∩ D, we define a local intersection number using local equations f_{p} = 0 for C and g_{p} = 0 for D at p:

(3.5)  (C · D)_{p} = dim_{ℂ} O_{S,p}/⟨f_{p}, g_{p}⟩.

Then the global intersection number of C and D is the sum of the local intersection numbers:

(3.6)  (C · D) = ∑_{p∈C∩D}(C · D)_{p}.

Example 3.5. For instance, say σ is any smooth cone in a complete smooth fan Σ. Let ρ and τ be the edges of σ, and let C = \overline{O(ρ)}, D = \overline{O(τ)} be the corresponding divisors on X_{Σ}. Then C ∩ D = {p_{σ}}, the T_{N}-fixed point corresponding to σ. Moreover by a change of basis in N, we can assume ρ = Cone(e_{1}) and τ = Cone(e_{2}), so C, D are defined in this coordinate system by the equations x = 0 and y = 0. Then (3.5) shows

(C · D)_{p_{σ}} = dim_{ℂ} ℂ[x, y]_{(0,0)}/⟨x, y⟩ = 1.

and hence (C · D) = 1.

On the other hand, if ρ and τ are not edges of the same two-dimensional cone, then in Exercise 3.5 you will show that (C · D) = 0. ⋄

Our goal is to extend this intersection product to a ℤ-valued bilinear form on the group of divisor classes on X_{Σ}, or Pic(X_{Σ}). The obstacle is that (3.6) does not apply when C = D, since then the set-theoretic intersection C ∩ D is not a finite set of points. To define (C · C), we must proceed in a different way. This topic is discussed in detail for general surfaces in [17, Chapter V] and in [30, Chapter VI].

For any curve C in a smooth surface S, we have the sheaf of ideals defining C,

I_{C} ≃ O(−C).

(This isomorphism comes from the fact that C defines a Cartier divisor.) The inclusion i : C ↪ S gives an exact sequence of sheaves

0 ⟶ I_{C} ⟶ O_{S} ⟶ i_{∗}O_{C} ⟶ 0,

where the sections of the sheaf i_{∗}O_{C} over U ⊆ S are i_{∗}O_{C}(U) = O_{C}(C ∩ U). The sheaf I_{C}/I_{C}^{2} is called the conormal sheaf of C in S and its dual

N_{C/S} = \underline{Hom}_{O_{S}}(I_{C}/I_{C}^{2}, O_{S})

is called the normal sheaf of C in S. In this situation, N_{C/S} is an invertible sheaf on C. Its sections can be interpreted as giving first-order deformations of C in S, so intuitively the degree of N_{C/S} on C measures the intersections of C and a “nearby” curve equivalent to C. This is the reason N_{C/S} will be used to define the intersection number (C · C).

In our situation on a smooth complete toric surface this can all be made very concrete. First, for each one-dimensional cone ρ, the divisor D_{ρ} satisfies

D_{ρ} ≃ ℙ^{1}

as a curve, and we have the standard degree isomorphism

deg : Pic(ℙ^{1}) ≃ ℤ

from (**ref**).

Definition 3.6. Let D_{ρ} be a divisor as above on the smooth toric surface X_{Σ}. The self-intersection number (D_{ρ} · D_{ρ}) is defined as

(D_{ρ} · D_{ρ}) = deg(N_{D_{ρ}/X_{Σ}}).

Moreover, these self-intersection numbers are determined by the fan Σ in the following way.

Theorem 3.7. Let D = D_{ρ} be the divisor on a smooth complete toric surface corresponding to ρ = Cone(v) which is a face of the smooth two-dimensional cones σ_{1} = Cone(u, v) and σ_{2} = Cone(v, w). Then

(D · D) = −b,

where u + w = bv as in (3.1).

Proof. We use e_{1} = w and e_{2} = v as our ℤ-basis for the lattice N. Then from (3.1),

u = av − w = −e_{1} + be_{2}.

Write x, y for the affine coordinates in U_{σ_{2}}. Then D is defined in U_{σ_{2}} by the equation y = 0. Similarly, D is defined in U_{σ_{1}} by the equation x^{b}y = 0. Thus the space of sections of I_{D}/I_{D}^{2} over U_{σ_{2}} ∩ D is one-dimensional and generated by y. Similarly x^{b}y generates the sections of I_{D}/I_{D}^{2} over U_{σ_{1}} ∩ D. Since x is the coordinate on D, this means that under the isomorphism D ≃ ℙ^{1},

I_{D}/I_{D}^{2} ≃ O_{ℙ^{1}}(b),

which has degree b. The normal sheaf N_{D/X_{Σ}} is the dual of this, so

deg(N_{D/X_{Σ}}) = −b,

which is what we wanted to show. □

Example 3.8. Let σ = Cone(v_{1}, v_{2}) be a smooth cone in a fan Σ′, and consider the star subdivision, in which Cone(v_{1} + v_{2}) is inserted to subdivide σ into two cones. Call the refined fan Σ. The exceptional divisor E of the blowup

π : X_{Σ} → X_{Σ′}

satisfies (E · E) = −1 on X_{Σ}. Curves with self-intersection number −1 on a surface are called exceptional curves of the first kind. They can always be contracted to a smooth point on a birationally equivalent surface, as in this toric example.

One of the foundational results in the theory of general algebraic surfaces is that every function field of transcendence degree 2 has a relatively minimal model, that is, a surface S such that if π : S → S′ is a birational morphism, then π is necessarily an isomorphism. The proof is given in [17, V.5.8]. Interestingly, the possible relatively minimal models for rational surfaces (ones whose function field is isomorphic to ℂ(x, y)) can be described in toric terms – they are precisely the surfaces ℙ^{2}, ℙ^{1} × ℙ^{1}, and H_{r} for r ≥ 2 from Theorem 3.3, and the proof of that theorem essentially shows that after contracting all possible exceptional curves of the first kind, every smooth toric surface will be taken to one of these.

Example 3.9. Consider a complete singular toric surface with a singular point p_{σ} corresponding to a cone σ with parameters m, 1 for some m > 1. If the singularity is resolved as in Example 1.8, consider the exceptional divisor E. Using the normalized basis of N relative to σ, the one-dimensional cone that is inserted in the refined fan is e_{1}, and

e_{2} + (me_{1} − e_{2}) = me_{1}.

Hence

(E · E) = −m

gives the self-intersection number of E. ⋄

The intersection product can now be extended by linearity to all of Pic(X_{Σ}), and this yields a ℤ-valued symmetric bilinear form.

Example 3.10. Consider the Hirzebruch surface H_{r}. We follow the notation from Example 3.4 above. We know that the group Pic(H_{r}) is generated by the classes of D_{1} and D_{2}. By Example 3.5 and Theorem 3.7, we have

(D_{1} · D_{1}) = −r, (D_{2} · D_{2}) = 0, (D_{1} · D_{2}) = (D_{2} · D_{1}) = 1.

The intersection product is described by the matrix

(\begin{matrix}(D_{1} · D_{1}) & (D_{1} · D_{2}) \\ (D_{2} · D_{1}) & (D_{2} · D_{2}) \end{matrix}) = (\begin{matrix}−r & 1 \\ 1 & 0 \end{matrix}) .

If D ≡ mD_{1} + nD_{2} and E ≡ pD_{1} + qD_{2} are any two divisors on the surface, then

(3.7)  (D · E) = (m n) (\begin{matrix}−r & 1 \\ 1 & 0 \end{matrix}) (\begin{matrix}p \\ q \end{matrix}) = −rmp + np + mq.

For instance, with D = E = D_{3} = D_{1} + rD_{2}, we obtain

(D_{3} · D_{3}) = −r + r + r = r.

The self-intersection number (D_{2} · D_{2}) = 0 is a reflection of the fibration structure on H_{r} studied in Example 3.15 from Chapter 3. The divisors D_{2} and D_{4} are fibers of the mapping H_{r} → ℙ^{1}. Such curves always have self-intersection equal to zero. You will compute several other intersection products on H_{r} in Exercise 3.6. ⋄

Now suppose the complete smooth toric surface X_{Σ} is obtained via a resolution of singularities of some normal toric surface containing a singularity p_{σ} corresponding to a two-dimensional cone σ with parameters m, k. From Theorem 2.7 and Theorem 3.7, if the Hirzebruch-Jung continued fraction expansion of m/k is

m/k = [[b_{1}, b_{2}, …, b_{r}]],

then the components of the exceptional divisor mapping to p_{σ}, which will be denoted D_{1}, …, D_{r}, satisfy

(3.8)  (D_{i} · D_{j}) = \begin{cases}−b_{i} & if j = i \\ 1 & if |i − j| = 1 \\ 0 & otherwise. \end{cases}

You will show in Exercise 3.7 that the matrix of the intersection product restricted to the subgroup of Pic(X_{Σ}) generated by the classes of the D_{i} defines a negative definite quadratic form. This condition is necessary for contractibility of a curve C on a smooth surface S, that is, existence of a birational morphism π : S → X, where π(C) is a (possibly singular) point on X.

Rational Double Points Reconsidered. If σ has parameters m, m − 1, then from Exercise 2.8, the Hirzebruch-Jung continued fraction expansion of m/(m − 1) is

m/(m − 1) = [[2, 2, …, 2]],

with m − 1 terms. Hence b_{i} = 2 for all i, and (3.8) gives the tridiagonal (m − 1) × (m − 1) matrix

( \begin{matrix}−2 & 1 & 0 & ⋯ & 0 \\ 1 & −2 & 1 & ⋯ & 0 \\ 0 & ... & ... & ... & 0 \\ 0 & & 1 & −2 & 1 \\ 0 & ⋯ & 0 & 1 & −2 \end{matrix} ) .

representing the intersection product on the subgroup of Pic(X_{Σ}) generated by the components of the exceptional divisor for the resolution of a rational double point of type A_{m−1}. At this point we can fully explain the terminology for these singularities.

The problem of classifying lattices

ℤe_{1} ⊕ ⋯ ⊕ ℤe_{s}

with negative definite bilinear forms B satisfying B(e_{i}, e_{i}) = −2 for all i arises in many areas within mathematics, most notably in the classification of complex simple Lie algebras via root systems. The matrix above is the (negative of) the Cartan matrix for the root system type A_{m−1}, which is often represented by the Dynkin diagram:

[figure]

with m − 1 vertices. The vertices represent the lattice basis vectors. The edges connect the pairs with B(e_{i}, e_{j}) ≠ 0 and B(e_{i}, e_{i}) = −2 for all i as above. In our case, the vertices represent the components D_{i} of the exceptional divisor, and the bilinear form is the intersection product.

A precise definition of a surface rational double point follows. We will say that a resolution of singularities

π : Y → X.

is minimal if no component of the exceptional divisor is a curve with self-intersection number −1 (see Example 3.8 and the following comments).

Definition 3.11. A singular point p of a surface X is said to be a rational double point, or Du Val singularity if there is a minimal resolution of singularities π : Y → X such that every irreducible component E_{i} of the exceptional divisor E over p, i = 1, …, r, satisfies

(3.9)  (K_{Y} · E_{i}) = 0,

where K_{Y} is the canonical divisor class on Y.

By the adjunction formula on Y (see [30, Chapter IV], [17, Chapter V], or §4 of this chapter), the equations (K_{Y} · E_{i}) = 0 for all i imply that E_{i} ≃ ℙ^{1} is a smooth curve, and (E_{i} · E_{i}) = −2 for all i. It follows that E = E_{1} + ⋯ + E_{r} also satisfies (E · E) = −2. It also follows from this condition that the canonical sheaf on Y is the pullback of the canonical sheaf on X under π, and this is one way to define rational singularities more generally. All toric surface singularities satisfy this last condition, but they may not be realizable as singularities of embedded surfaces in ℂ^{3} , so we may not be able to define a multiplicity in the naive way we did in Example 1.6. In Exercise 3.8, you will verify some of these claims.

Exercises for § 3.

3.1. In this exercise, you will verify several statements made in the proof of Lemma 3.2 (a) Show that if the cone σ is strictly convex, then the integers a_{i} in (3.1) must be strictly

positive.

(b) Show that if v_{i−1} + v_{i+1} = v_{i}, then Cone(v_{i−1}, v_{i+1}) must also be smooth.

3.2. In the proof of Theorem 3.3, verify that if v_{j} = −v_{1} and v_{0} = −v_{2} + a_{1}v_{1} with a_{1} < −2, then Σ is a refinement of a fan Σ′ with X_{Σ′} ≃ H_{r}, where r = |a_{1}|.

3.3. In this exercise, you will supply details for several claims in the last paragraph of the proof of Theorem 3.3.

(a) Show that v_{j} + v_{j+1} must lie in the third quadrant when v_{j} is the last vector in the fourth quadrant, and v_{j+1} lies in the second quadrant.

(b) Verify the details to show that bc − ad > 1 in both cases d > 0 and d < 0.

3.4. In the statement of Theorem 3.3, you might have noticed the absence of the Hirzebruch surface H_{1}. Show that this surface is actually isomorphic to the blowup of ℙ^{2} at one T_{N}- fixed point.

3.5. Show that if C, D are the divisors on X_{Σ} corresponding to one-dimensional cones that are not faces of the same two-dimensional cone, then

(C · D) = 0.

3.6. This exercise studies several further examples of the intersection product on H_{r}.

(a) Check the equation (D_{3} · D_{3}) = r using Theorem 3.7.

(b) Compute (D_{4} · D_{4}) using (3.7) and also directly from Theorem 3.7.

(c) Compute K^{2} = (K · K) on H_{r}.

3.7. Use (3.8) to show that the restriction of the intersection product to the subgroup of Pic(X_{Σ}) generated by the components of the exceptional divisor of the resolution of a singular point p_{σ} corresponding to a cone σ with parameters m, k yields a negative-definite associated quadratic form. Hint: recall that if B(x, y) is a bilinear form, the associated quadratic form is Q(x) = B(x, x).

3.8. This exercise deals with the rational singularities from Definition 3.11.

(a) Show that (3.9)is satisfied for the standard toric resolution of the rational double point of type A_{m−1}.

(b) Show that E = E_{1} + ⋯ + E_{m−1} on the toric resolution satisfies (E · E) = −2.

(c) Show that if b_{i} ≥ 3 for some i in the Hirzebruch-Jung continued fraction expansion of m/k, then the components of the exceptional divisor on the toric resolution do not satisfy (3.9).

(d) (This part requires use of the adjunction formula – see §4 of this chapter.) Show that if C is an irreducible curve on a smooth surface Y with (C ·C) < −1 and (K_{Y} ·C) = 0, then C must be isomorphic to ℙ^{1}.

3.9. Another interesting numerical fact about the integers b_{i} from (3.1) is the following. Suppose a smooth fan Σ has one-dimensional cones labeled as in Lemma 3.1. Then

(3.10)  b_{0} + b_{1} + ⋯ + b_{d−1} = 3d − 12.

This exercise will sketch a proof of this fact.

(a) Show that (3.10) holds for the standard fans yielding ℙ^{2}, ℙ^{1} × ℙ^{1}, and H_{r}, for r ≥ 2.

(b) Show that if (3.10) holds for a smooth Σ, then it holds for the fan obtained by performing a star subdivision on one of the two-dimensional cones of Σ.

(c) Deduce that (3.10) holds for all smooth fans using Theorem 3.3.

4. Riemann-Roch and Lattice Polygons

Riemann-Roch theorems are a class of results about the dimensions of sheaf cohomology groups. The original statement along these lines was the theorem of Riemann and Roch concerning sections of line bundles on algebraic curves. This result and its generalizations to higher-dimensional varieties can be formulated most conveniently in terms of the following notion of the Euler characteristic of a sheaf.

Definition 4.1. Let F be a coherent sheaf on a projective variety X. The Euler characteristic of F, denoted χ(F) is the integer

χ(F) = ∑_{i=0}^{dim(X)}(−1)^{i} dim H^{i}(X, F).

For instance, if X is a curve and L is a line bundle on X, then

χ(L) = dim H^{0}(X, L) − dim H^{1}(X, L).

One of the key facts we will need about Euler characteristics is an additivity statement.

Proposition 4.2. Let

(4.1)  0 ⟶ F′ ⟶ F ⟶ F′′ ⟶ 0

be a short exact sequence of coherent sheaves on a projective variety X. Then

χ(F) = χ(F′) + χ(F′′).

Proof. This is a consequence of the long exact cohomology sequence associated to (4.1). You will supply the details in Exercise 4.1. □

Riemann-Roch Theorem for a Curve. A modern form of the original Riemann-Roch theorem for curves is the claim that if L = O_{C}(D) is the line bundle corresponding to the divisor D on a smooth projective curve C, then

(4.2)  χ(O_{C}(D)) = deg(D) + χ(O_{C}).

This equality is usually rewritten using Serre duality as in Chapter 7. Namely, if K is a canonical divisor on C, then we have

H^{1}(O_{C}(D)) ≃ H^{0}(O_{C}(K − D))^{∗}
H^{1}(O_{C}) ≃ H^{0}(O_{C}(K))^{∗}.

We will call the integer g = dim H^{0}(O_{C}(K)) the genus of the curve C. Then (4.2) can be rewritten in the form commonly used in the theory of curves:

(4.3)  dim H^{0}(O_{C}(D)) − dim H^{0}(O_{C}(K − D)) = deg(D) + 1 − g.

A proof of this theorem and a number of its applications are given in [17, Chapter IV]. Also see Exercise 4.4 below. As a first consequence, note that if D = K_{C} is a canonical divisor, then

(4.4)  deg(K_{C}) = 2g − 2.

We will need to use (4.3) most often in the simple case X ≃ ℙ^{1}. Then g = 0 and the Riemann-Roch theorem for ℙ^{1} is the statement

(4.5)  χ(O_{ℙ^{1}}(D)) = deg(D) + 1

for all divisors D on ℙ^{1}.

The Adjunction Formula. For a curve C in a smooth surface X, we also have a relation between the canonical divisors on the curve and the surface. From the residue exact sequence from Chapter 7, if K_{X} is the canonical sheaf on the surface and ω_{C} is the canonical sheaf on the curve (or the dualizing sheaf if C is singular), then

0 ⟶ K_{X} ⟶ K_{X} ⊗ O_{X}(C) ⟶ ω_{C} ⟶ 0.

In other words, the sheaf ω_{C} on C is given by

(4.6)  ω_{C} ≃ K_{X} ⊗ O_{X}(C) ⊗ O_{C}.

Combining this with (4.4) above gives the following formula.

Theorem 4.3 (Adjunction Formula). Let C be a smooth curve in a smooth surface X, then in the intersection product on X,

(K_{X} · C) + (C · C) = 2g − 2,

where g is the genus of the curve C.

When C is a singular curve in a smooth surface, there is a similar formula, but the arithmetic genus p_{a} replaces the ordinary genus g. By definition,

p_{a} = (−1)(χ(O_{C}) − 1)

for a curve C. You will supply the details of the proof in Exercise 4.2.

Riemann-Roch Theorem for a Surface. The statement for surfaces corresponding to (4.2) is given next.

Theorem 4.4 (Riemann-Roch for Surfaces). Let X be a smooth projective surface, let O_{X}(D) be the line bundle on X corresponding to a divisor D, and let K be a canonical divisor. Then

(4.7)  χ(O_{X}(D)) = \frac{(D · D) − (D · K)}{2} + χ(O_{X}),

where (D · D) and (D · K) are the intersection products on X as in §3.

We will only prove this for X a smooth complete toric surface; there is a simple and concrete proof in this case.

Proof. Note that (4.7) certainly holds if D ≡ 0, so O_{X}(D) ≃ O_{X}. Our proof will use the special properties of smooth X toric surfaces in the following way. Recall that if X = X_{Σ}, then Pic(X) is generated by the isomorphism classes of O(D_{i}) where D_{i}, i = 1, …, d are the divisors corresponding to the one-dimensional cones in Σ. Hence, to show that (4.7) holds for all D on X, by a “two-sided induction,” it suffices to show that if (4.7) holds for a divisor D, then it also holds for D′ = D + D_{i} and D′′ = D − D_{i} for all i.

Assume first that (4.7) holds for D, and consider D′ = D + D_{i}. From the exact sequence

0 ⟶ O_{X}(−D_{i}) ⟶ O_{X} ⟶ O_{D_{i}} ⟶ 0,

if we take tensor product with O_{X}(D+D_{i}) we obtain an exact sequence of sheaves

0 ⟶ O_{X}(D) ⟶ O_{X}(D + D_{i}) ⟶ O_{D_{i}}(D + D_{i}) ⟶ 0.

By Proposition 4.2, it follows that

χ(O_{X}(D + D_{i})) = χ(O_{X}(D)) + χ(O_{D_{i}}(D + D_{i})).

By the induction hypothesis, the first term on the right is

(4.8)  χ(O_{X}(D)) = \frac{(D · D) − (K · D)}{2} + χ(O_{X}).

The second term can be computed using the Riemann-Roch theorem for curves and the fact that D_{i} ≃ ℙ^{1}. By (4.5),

(4.9)  χ(O_{D_{i}}(D + D_{i})) = (D · D_{i}) + (D_{i} · D_{i}) + 1.
Hence combining (4.8) and (4.9), we have χ(O_{X}(D + D_{i}))
= \frac{(D · D) − (K · D)}{2} + (D · D_{i}) + (D_{i} · D_{i}) + 1 + χ(O_{X})
= \frac{((D + D_{i}) · (D + D_{i})) + (D_{i} · D_{i}) − (K · D) + 2}{2} + χ(O_{X})
= \frac{((D + D_{i}) · (D + D_{i})) − (K · (D + D_{i}))}{2} + χ(O_{X}),

where the last equality follows since K = −(D_{1} + ⋯ + D_{d}), hence

(K · D_{i}) = −(D_{i} · D_{i}) − 2.

Recall that (D_{i} · D_{j}) is 1 when the corresponding one-dimensional cones in Σ are faces of the same two-dimensional cone and 0 otherwise.

The case D′′ = D − D_{i} is similar and is left to the reader as Exercise 4.3. □

By the results of Chapter 7, for a smooth toric surface X we have

χ(O_{X}) = dim H^{0}(O_{X}) − dim H^{1}(O_{X}) + dim H^{2}(O_{X}) = 1 − 0 + 0.

Moreover, in the case that D = ∑ m_{i}D_{i} with m_{i} ≥ 0 for all i, it will be true that

dim H^{1}(O_{X}(D)) = dim H^{2}(O_{X}(D)) = 0.

Hence for toric surfaces, Theorem 4.4 gives even more direct information about dim H^{0}(O_{X}(D)) than might be apparent at first glance: For D ≡ ∑ m_{i}D_{i} with m_{i} ≥ 0 for all i,

(4.10)  dim H^{0}(O_{X}(D)) = \frac{(D · D) − (K · D)}{2} + 1,

and the terms on the right can all be computed explicitly from the geometry of the fan Σ. We will consider an example shortly. The following statement is sometimes considered a part of the Riemann-Roch Theorem for surfaces.

Theorem 4.5 (Noether’s Theorem). Let X be a smooth surface, and let K_{X} be a canonical divisor, then

χ(O_{X}) = \frac{(K · K) + e(X)}{12},

where e(X) is the topological Euler characteristic of X as a four-dimensional real manifold.

The Riemann-Roch theorems given so far have been vastly generalized by results of of Hirzebruch and Grothendieck, and the precise relation of Noether’s Theorem to the Riemann-Roch Theorem for surfaces is a special case of their approach. Hence we will postpone considering this until Chapter 11.

Lattice Polygons. For the remainder of this section, we will indicate some of the ways that geometric results about toric surfaces studied so far in this section are related to geometry and combinatorics of lattice polygons – polygons which are the convex hull of a finite number of integer lattice points in the plane. The connection comes from the construction of a projective toric variety from a polytope from Chapter 2.

Example 4.6. Let X be the Hirzebruch surface H_{2}. We will use the notation from Example 3.4 and Example 3.10. Let

D = D_{1} + 4D_{2}.

Then since K ≡ −2D_{1} − 4D_{2}, it is easy to compute that

(D · D) = 6, (D · K) = −8.

Hence from (4.10),

dim H^{0}(O_{H_{2}} (D)) = \frac{6 + 8}{2} + 1 = 8.

From Chapter 6, §2, recall that divisors on H_{2} can also be defined via polygons

Figure 4. The polygon corresponding to the divisor D and its normal fan.

that are compatible with the standard fan Σ yielding the Hirzebruch surface. Writing v_{i} for the ray generators of the one-dimensional cones,

v_{1} = e_{2}, v_{2} = e_{1}, v_{3} = −e_{2}, v_{4} = −e_{1} + 2e_{2},

we claim that this D comes from the polygon P_{D} in M_{ℝ} shown in Figure 4 and defined by the facet presentation

⟨u, v_{1}⟩ ≥ 0
⟨u, v_{2}⟩ ≥ 0
⟨u, v_{3}⟩ ≥ −1
⟨u, v_{4}⟩ ≥ −2.

By the general results of Chapter 6,

D ≡ 0 · D_{1} + 0 · D_{2} + 1 · D_{3} + 2 · D_{4} ≡ D_{1} + 4D_{2},

where the last equivalence comes from the facts that D_{4} ≡ D_{2} and D_{3} ≡ D_{1} + 2D_{2}. This establishes our claim.

This is the same as the polytope defining the rational scroll from Example 3.15 in Chapter 2. Note that

dim H^{0}(O_{H_{2}} (D)) = 8 = |P_{D} ∩ M|,

the number of lattice points in P_{D}. ⋄

In fact, the same toric set-up may be adapted to study lattice points in any lattice polygon. Let P ⊆ M_{ℝ} ≃ ℝ^{2} be any lattice polygon (two-dimensional lattice polytope). By the results of Chapter 2, §3, we know that there is an associated projective toric surface X_{P}. As abstract varieties

X_{P} ≃ X_{Σ_{P}} ,

where Σ_{P} is the normal fan of P in the dual space N_{ℝ}. If Σ_{P} is not a smooth fan, then we can refine it as in Theorem 1.10 to obtain a resolution of singularities π : Y → X_{P}. We will work on the smooth surface Y if X_{P} is not smooth in the following.

From Chapter 6, the polygon P also determines a line bundle O_{Y} (D_{P}) and divisor (class) D_{P} on X_{P} or Y. Moreover, by (** ref **),

H^{0}(O_{Y} (D_{P})) = ⊕_{u∈P ∩M} ℂ · χ^{u}.

If we combine this with the Riemann-Roch equality (4.10), then we have a formula relating the number of lattice points in the polygon P to quantities from the geometry of the associated toric surface Y:

(4.11)  |P ∩ M| = \frac{(D_{P} · D_{P}) − (K_{Y} · D_{P})}{2} + 1.

In fact, both (D_{P} · D_{P}) and (−K_{Y} · D_{P}) also have elementary “polytopal” interpretations in terms of the geometry of the polygon P.

We begin by noting that if P is given by the facet presentation

⟨u, v_{i}⟩ ≥ −a_{i},

using the primitive integer inward normals v_{i} for each face, then the inequalities

(4.12)  ⟨u, v_{i}⟩ ≥ −a_{i} + 1

define a second polygon Q which is the convex hull of the lattice points in the interior of P. You will prove this in Exercise 4.5. If P comes from the divisor

D = a_{1}D_{1} + ⋯ + a_{d}D_{d}

on Y, then since

K_{Y} ≡ −(D_{1} + ⋯ + D_{d}),

the inequalities (4.12) show the first statement in the following proposition. We let Int(P) denote the interior of P.

Proposition 4.7. With notation as above,

|Int(P) ∩ M| = dim H^{0}(O_{Y} (D + K_{Y})).

Moreover, when D + K_{Y} is equivalent to an effective divisor,

|Int(P) ∩ M| = \frac{(D · D) + (K_{Y} · D)}{2} + 1.

Proof. The first equality follows from (4.12). The second is an easy consequence of the Riemann-Roch formula (4.10). You will check the details in Exercise 4.6.

□

This proposition also has an interesting connection with the adjunction formula (Theorem 4.3). If we look at the embedded projective surface Y ⊆ ℙ^{m} corresponding to P, then D represents the hyperplane section class on Y, that is the divisor class containing the curves Y ∩ H for hyperplanes H ⊆ ℙ^{m}. Hence |Int(P) ∩ M| gives what is known as the sectional genus of the surface.

Example 4.8. Let P = Δ_{n} = Conv({ne_{1}, ne_{2}, 0}). Then Y ≃ ℙ^{2} in its nth Veronese embedding and D_{P} ≡ nL, where L is the class of a line. The hyperplane sections are the curves of degree n in ℙ^{2}, which all have (arithmetic) genus

|Int(Δ_{n}) ∩ M| = \frac{(n − 1)(n − 2)}{2}.

You will check this assertion and another example in Exercise 4.7. ⋄

We now return to the promised polytopal interpretations of the terms on the right in (4.11). Our next result gives the polytopal interpretation of the intersection product (−K_{Y} · D_{P}). Let ∂P denote the boundary of P in M_{ℝ} ≃ ℝ^{2}.

Proposition 4.9. Let D_{P} be the effective divisor corresponding to P. Then

(−K_{Y} · D_{P}) = |∂P ∩ M|.

Proof. The elementary formula

|P ∩ M| = |Int(P) ∩ M| + |∂P ∩ M|

is clear. Substituting from (4.10) and Proposition 4.7, this becomes

\frac{(D_{P} · D_{P}) − (K_{Y} · D_{P})}{2} + 1 = \frac{(D_{P} · D_{P}) + (K_{Y} · D_{P})}{2} + 1 + |∂P ∩ M|.

Some simple algebraic manipulation yields the equation to be proved. □

The other term on the right in (4.11) is (D_{P} · D_{P})/2. This can be interpreted using the machinery of BKK theory as in [8, Chapter 7, Theorem (5.4)]. Recall that if f, g are two Laurent polynomials in ℂ[x, x^{−1}, y, y^{−1}], provided that |V(f)∩ V(g)∩(ℂ^{∗})^{2}| is finite, there is an upper bound on this number which is determined by the geometry of the Newton polygons P = NP(f) and Q = NP(g). Namely

|V(f) ∩ V(g) ∩ (ℂ^{∗})^{2}| ≤ area(P + Q) − area(P) − area(Q),

where P + Q is the Minkowski sum. Moreover, the inequality will be an equality for generic choices of f, g with these Newton polygons. In our case if we take two general elements f, g of ⊕_{u∈P ∩M}ℂ · χ^{u}, then Q = P as well, and the two curves will intersect only in a finite number of points contained in (ℂ^{∗})^{2}. The intersection number will be

(D_{P} · D_{P}) = area(2P) − area(P) − area(P) = 2 area(P).

This shows the following statement.

Proposition 4.10. Let D_{P} be the divisor on the smooth toric surface Y corresponding to the lattice polygon P. Then

(D_{P} · D_{P}) = 2 area(P).

We are now ready to reformulate the equation (4.11). Substituting from Proposition 4.9 and Proposition 4.10, we obtain

(4.13)  |P ∩ M| = area(P) + \frac{|∂P ∩ M|}{2} + 1.

This is actually a very well-known result in the theory of lattice polygons called Pick’s Theorem. Our argument leading to (4.13) via the Riemann-Roch Theorem is certainly not the most elementary way of proving this statement. But it provides an interesting glimpse of the way toric surfaces and lattice polygons are related.

Reflexive Polygons and The Number 12. The final topic we will discuss gives another example of an interaction between toric geometry and lattice polygons, following the article [27]. Recall from Definition 3.11 of Chapter 2 that a lattice polytope in N_{ℝ} is said to be reflexive if its facet presentation is

P = {m ∈ N_{ℝ} | ⟨m, u_{F} ⟩ ≥ −1 for all facets F }.

It follows that if P is reflexive, the origin is the unique interior lattice point of P (Exercise 3.5 of Chapter 2). Since a_{F} = 1 for all F, Exercise 2.1 of Chapter 2 implies that the dual polytope is

P^{∘} = Conv(u_{F} | F facet of P).

Thus P^{∘} is a lattice polytope and is also reflexive (Exercise 3.5 of Chapter 2).

Up to the action of matrices in GL(2, ℤ) and translations by elements of the lattice, there are exactly 16 equivalence classes of reflexive polygons in the plane, shown in Figure 5.

The article [27] gives four different proofs of the following result.

Theorem 4.11. Let P be a reflexive lattice polygon in N_{ℝ} containing the origin. Then

|∂P ∩ N| + |∂P^{∘} ∩ M| = 12.

One of these proofs consists of a case-by-case verification of the statement for each of 16 equivalence classes in Figure 5. You will prove the theorem this way in Exercise 4.8. This approach is straightforward, but not very enlightening! We will give two other proofs, both using geometry of toric surfaces.

Proof. For the first, we will need the following lemma, which describes a way to construct a fan Σ(P), different from the normal fan, from each of these polygons.

Lemma 4.12. Let P be a reflexive polygon with the origin as its unique interior lattice point. Let Σ(P) be the complete fan whose one-dimensional cones are spanned by the rays from the origin to the points on the boundary of P. Then Σ(P) is a smooth fan.

Figure 5. The 16 equivalence classes of reflexive lattice polygons (from [27])

This is easy to see for each of the polygons in Figure 5. Also see Exercise 4.9. Now let Y (P) be the smooth toric surface corresponding to Σ(P), and let

D_{1}, …, D_{d} be the divisors corresponding to the one-dimensional cones with ray generators v_{1}, …, v_{d}. Note that

d = |∂P ∩ N|.

Consider the anticanonical divisor

−K_{Y (P)} ≡ D_{1} + ⋯ + D_{d}

on Y. The polygon in M_{ℝ} corresponding to this divisor has facet representation,

Q = {m ∈ M_{ℝ} : ⟨m, v_{i}⟩ ≥ −1 for all i}.

However, since P is the convex hull of the v_{i}, this means that Q = P^{∘} is the dual reflexive polygon. Hence −K_{Y (P)} is ample. Varieties with this property are called Fano varieties. Smooth Fano surfaces are also called Del Pezzo surfaces.

Let Y = Y (P) for one of the reflexive polygons in the plane. By Proposition 4.9, taking D = −K_{Y}, we have

(4.14)  (−K_{Y} · −K_{Y}) = |∂P^{∘} ∩ M|.

However, we can also compute (−K_{Y} · −K_{Y}) directly using the intersection products

(D_{i} · D_{j}) = \begin{cases}−b_{i} & if i = j \\ 1 & if v_{i} and v_{j} are adjacent \\ 0 & otherwise, \end{cases}

where the b_{i} are the integers from Lemma 3.1. This computation shows

(4.15)  (−K_{Y} · −K_{Y}) = − ∑_{i=1}^{d} b_{i} + 2d = 12 − d,

where the last equation follows from Exercise 3.9. Since d = |∂P ∩N|, combining (4.14) and (4.15) yields

|∂P ∩ N| + |∂P^{∘} ∩ M| = 12,

which is what we wanted to show. □

The second proof we will present is even simpler, though less elementary.

Proof. We construct the smooth toric surface Y exactly as in the first proof. By Noether’s Theorem quoted above (Theorem 4.5),

χ(O_{Y}) = 1 = \frac{(K_{Y} · K_{Y}) + χ(Y)}{12}

for a smooth toric surface, since H^{1}(O_{Y}) = H^{2}(O_{Y}) = 0. From (4.14) above,

(K_{Y} · K_{Y}) = |∂P^{∘} ∩ M|.

The topological Euler characteristic χ(Y) is the alternating sum of the Betti numbers (ranks of the singular homology groups with integer coefficients). But Y is connected, orientable, simply-connected, and satisfies Poincaré duality. Hence

χ(Y) = 1 − 0 + (d − 2) − 0 + 1 = d = |∂P ∩ N|.

Hence the desired result follows directly from Noether’s Theorem. □

Higher dimensional reflexive polytopes and their associated toric varieties are currently being actively studied. See for instance [25]. One reason for the interest in these varieties is the relation with mirror symmetry in physics. (**need more complete summary of literature, pointer ahead to where this discussion will be continued (if it will!)**)

4. Riemann-Roch and Lattice Polygons 185

Exercises for § 4.

4.1. Use the long exact cohomology sequence associated to (4.1) to prove Proposition 4.2.

4.2. Prove the Adjuction Formula, Theorem 4.3, using (4.6) and (4.4).

4.3. Complete the proof of Theorem 4.4 by showing that if (4.7) hold for D, then it also holds for D′′ = D − D_{i} where D_{i} is any one of the divisors corresponding to the onedimensional cones in Σ.

4.4. The Riemann-Roch theorem for curves, in the form (4.2), can be proved by much the same method as used in the previous exercise. Namely, show that if (4.2) holds for a divisor D then it also holds for the divisors D + P and D − P, where P is an arbitrary point on the curve.

4.5. Let P ⊆ M_{ℝ} be the polygon defined a collection of inequalities

⟨u, v_{i}⟩ ≥ −a_{i}, 1 ≤ i ≤ d,

where the v_{i} are a collection of primitive integer vectors in M. Show that the inequalities

⟨u, v_{i}⟩ ≥ −a_{i} + 1, 1 ≤ i ≤ d,

define the convex hull of the interior lattice points in P.

4.6. Prove the second assertion in Proposition 4.7 using the Riemann-Roch theorem.

4.7. This exercise studies the sectional genus of toric surfaces introduced in Example 4.8.

(a) Verify the formula given in the example for the sectional genus of ℙ^{2} in its nth Veronese embedding.

(b) Let P = Conv({0, ae_{1}, be_{2}, ae_{1} + be_{2}}). What is the abstract smooth toric surface Y in this case? Show that its sectional genus is (a − 1)(b − 1).

4.8. This exercise develops a brute-force proof of Theorem 4.11.

(a) For each of the polygons in Figure 5, determine the dual polygon P^{∘}. (Some are self-dual.)

(b) Using your results from part a, verify the equality |∂P ∩N|+|∂P^{∘}∩M| = 12 directly.

4.9. In this exercise, you will show a criterion that implies the fan from Lemma 4.12 is smooth.

(a) Show that a two-dimensional cone σ = Cone(v_{1}, v_{2}) in N is smooth if and only if Conv(0, v_{1}, v_{2}) ∩ N = {0, v_{1}, v_{2}}. Hint: Show that if e_{1}, e_{2} is a normalized basis of N with respect to σ, then if m > 1, then e_{1} is in Conv(0, v_{1}, v_{2}) ∩ N.

(b) Deduce that the fan Σ from Lemma 4.12 is smooth whenever P is reflexive.

4.10. According to Theorem 3.3, every smooth toric surface is a blow-up of either ℙ^{2}, ℙ^{1} × ℙ^{1}, or H_{r} for r ≥ 2.

(a) For each of the 16 reflexive polygons in Figure 5, we have a smooth toric surface Y (P) associated to the fan Σ(P) as in Lemma 4.12. Where does Y (P) fit in this classification in each case? (This gives a classification of toric Del Pezzo surfaces.)

(b) From each of the polygons in Figure 5, we could also construct a fan Σ′(P) by taking the rays from the interior lattice point at the origin through vertices (but not the other lattice points on the boundary). How many of the surfaces X_{Σ′(P)} are smooth? Identify the ones that are.

4.11. From Chapter 7, §3, recall that the Erhart polynomial of a lattice polytope P ⊂ M is given by

L_{P} (k) = |kP ∩ M|,

where kP = P + ⋯ + P (Minkowski sum). Show that if P is a lattice polygon in the plane, then

L_{P} (k) = c_{2}k^{2} + c_{1}k + c_{0},

for certain rational numbers c_{2}, c_{1}, c_{0}. Indentify these coefficients using the geometry of P.

Chapter 9

Resolution of Singularities

Part III

Advanced Theory of Toric Varieties

Chapters 10 to 13 explore further aspects of the theory of toric varieties. This part of the book assumes a wide knowledge of algebraic geometry, though we give careful references for all of the tools used in our study of toric geometry. The topics presented here are just of few of many rich areas of inquiry encountered in the study of toric varieties.

Chapter 10

Geometric Invariant Theory

Chapter 11

The Riemann-Roch Theorem

Chapter 12

The Topology of Toric Varieties

Chapter 13

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