Math in the Mountains Toric Varieties Tutorial July 29 - 31, 2013

Exercises on Toric Varieties

1. Show that the rational quartic in P^{3} corresponding to A = {(4, 0), (3, 1), (1, 3), (0, 4)} is not projectively normal (Harshorne, I.3.18)

2. Show that the tetrahedron P = conv{0, e_{1}, e_{2}, e_{1} + e_{2} + 2e_{3}} ⊂ R^{3} is not normal. (A lattice polytope P ⊂ M_{R} is normal if (kP ∩ M) + (ℓP ∩ M) = (k + ℓ)P ∩ M for all k, ℓ ∈ N.)

3. Show that the complete fan Σ in N_{R} ≃ R^{2} with rays R_{≥0}e_{1}, R_{≥0}e_{2}, and R_{≥0}(−e_{1}−e_{2}) gives X_{Σ} = P^{2}.

4. Identify the toric surface given by the complete fans with rays ±R_{≥0}e_{1}, ±R_{≥0}e_{2} (four 2-dimensional cones)

5. Identify the toric surface given by the complete fans with rays R_{≥0}e_{1}, ±R_{≥0}e_{2}, and ±R_{≥0}(−e_{1} + ae_{2}) where a ≥ 1 (four 2-dimensional cones)

6. Show using the toric description that Cl(P^{1}) = Z.

7. Complete the proof that we have an exact sequence for the class group of X_{Σ}:

M → Z^{|Σ(1)|} → Cl(X_{Σ}) → 0,

where the first map is given by the integer matrix whose rows are the primitive generators of the rays of Σ.

8. (Toric surfaces, Riemann-Roch, etc.) The Riemann-Roch theorem on a smooth algebraic surface X can be written as

(1)  χ(O_{X}(D)) = \frac{D(D − K_{X})}{2} + χ(O_{X}),

where D is a divisor on X, K_{X} is the canonical divisor, and

χ(F) = h^{0}(F) − h^{1}(F) + h^{2}(F)

is the Euler characteristic (there would be additional terms if dim X > 2).

a. Given a lattice polytope P in Z^{2}, show how to use the facet description of P to

find D and Σ such that a basis for H^{0}(O_{X_{Σ}}(D)) is given by P ∩ Z^{2}. b. Show that in this case χ(O_{X_{Σ}}(D)) = |P ∩ Z^{2}| and χ(O_{X_{Σ}}) = 1 (Hint: Demazure

+ support function.) c. Replace D with nD in the formula (1) above to get

h^{0}(O_{X_{Σ}}(nD)) = \frac{D^{2}}{2} n^{2} − \frac{DK}{2} n + 1

d. Pick’s formula for a lattice polygon implies

|nP ∩ Z^{2}| = Area(P)n^{2} + \frac{|∂P ∩ Z^{2}|}{2}n + 1.

Check this for the triangle P = conv{(0, 0), (2, 0), (0, 2)}. e. Conclude: We can compute D^{2} and DK from purely combinatorial data from P

for a toric surface. In fact K_{X_{Σ}} = − ∑_{ρ∈Σ(1)} D_{ρ} when X_{Σ} is smooth.

Exercises on Algebraic Statistics Topics

1. Give an integer matrix A such that the corresponding toric model coincides with the 3 × 3 independence model. Generalize to obtain a matrix corresponding to the k × ℓ independence model for all k, ℓ ≥ 2.

2. For the Jukes-Cantor model,

a. Compute P (ACG), P (AAC), P (ACA) and P (CAA). Explain where the p_{dis}, p_{ij} polynomials come from and why every component of the full Jukes-Cantor model can be expressed using just these 5 distinct polynomials.

b. Verify that p_{123} + p_{dis} + p_{12} + p_{13} + p_{23} = 1. c. Show that

\begin{aligned}q_{111} & = (θ_{1} − π_{1})(θ_{2} − π_{2})(θ_{3} − π_{3}) \\ q_{110} & = (θ_{1} − π_{1})(θ_{2} − π_{2})(θ_{3} + 3π_{3}) \\ q_{101} & = (θ_{1} − π_{1})(θ_{2} + 3π_{2})(θ_{3} − π_{3}) \\ q_{011} & = (θ_{1} + 3π_{1})(θ_{2} − π_{2})(θ_{3} − π_{3}) \\ q_{000} & = (θ_{1} + 3π_{1})(θ_{2} + 3π_{2})(θ_{3} + 3π_{3}) \end{aligned}

are linear combinations of p_{123}, p_{dis}, p_{ij} (and monomials in linear combinations of the original parameters).

d. What is the corresponding toric variety for this reparametrized model?

3. Let A be any integer matrix with equal column sums. Let φ be the parametrization of the corresponding toric model. Write p̂ for φ(θ̂), where θ̂ is the MLE for θ. Show using the method of Lagrange multipliers that if L is the likelihood function for given data u, then

L · b = L · Au = λ · Ap̂

Exercises on Geometric Modeling Topics

1. Show that for all choices of the control points P_{i} ∈ R^{2}, the Bézier cubic curve

b(t) = (x(t), y(t)) = ∑_{i=0}^{3} \binom{3}{i}t^{i}(1 − t)^{3−i}P_{i}

lies on an algebraic curve defined by an equation F (x, y) = 0, where F has total degree 3 in x, y. The curve C = V (F) is automatically rational (g(C) = 0) because of this. Explain why this implies C must have a singular point (an ordinary node or cusp), possibly with coordinates in C.

2. Some questions about toric surface patches.

a. Determine the Krasauskas toric blending functions for the triangle

Δ_{k} = conv{(0, 0), (k, 0), (0, k)}

and show that the corresponding surface patch is a reparametrization of the Bézier triangle patch of order k.

b. Now consider the Krasauskas toric blending functions for the triangle

Δ = conv{(0, 0), (2, 0), (1, 1)}

What happens on the image of a toric surface patch for this Δ at the corner (1, 1)?

c. Let P be a vertex of Δ. The “corner triangle” of P is the triangle consisting of P and the nearest two lattice points on the edges adjacent to P. One of the results of [K] is that the image of P on the toric surface patch is nonsingular if and only if the corner triangle has area 1/2 (equivalently, the two vectors from P to the nearest lattice points on the adjacent edges form a lattice basis for Z^{2}). Prove this statement and identify the singularity if the corner triangle has area ℓ/2 for ℓ > 1.

3. Let P be a set of control points in R^{3}. Show that the degree of the implicit equation defining an algebraic surface containing the image of b_{A,w,P} is at most

2 · Area(conv(A))

and equals this for generic choices of P. (Hint: Use the BKK bound.)

4. What is the relation between the image of b_{A,w,P} and the image of b_{A′,w,P′} if A′ ⊂ A and P′ is the subset of P corresponding to A′?

5. Let Δ be a lattice polygon in the plane, let h_{m} be the equations of the edges using inward normals, and let H : Δ → R^{ℓ} be the map defined by the h_{i} The toric blending functions come from composing this H with χ : R^{ℓ} → R^{ℓ} defined by

y ⟼ (y_{1}^{h_{1}(m)} ⋯ y_{ℓ}^{h_{ℓ}(m)} : m ∈ A)

The toric surface patch is the composition π_{P} ∘ χ ∘ H, where π_{P} : R^{ℓ} → R^{3} is the affine form of a linear projection defined by the set of control points Show that the m-component of χ(y) is just y^{a}z^{m}, where a = (a_{1}, …, a_{ℓ}) comes from the constant terms, and z = (z_{j}) where

z_{j} = ∏_{i=1}^{ℓ} y_{i}^{⟨v_{i},e_{j}⟩}, j = 1, 2

6. Show that the toric suface patch for the triangle conv{(0, 0), (k, 0), (0, k)} has linear precision.

7. Exactly how does the set-up for Birch’s Theorem relate to the algebraic moment map for Y_{A}?