\chapter{Projective Toric Varieties}
\label{poly}

\setcounter{section}{-1}

\section{Background:\ Projective Varieties}

Our discussion assumes that the reader is familiar with the elementary
theory of projective varieties, at the level of \cite[Chapter 8]{iva}.

In Chapter~\ref{affine}, we introduced affine toric varieties.  In
general, a toric variety is an irreducible variety $X$ over $\C$
containing a torus $T_N \simeq \torus$ as a Zariski open set such that
the action of $\torus$ on itself extends to an action on $X$. 
We will learn in Chapter~\ref{normal} 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
\begin{equation}
\label{2.pndef}
\P^{\,n} = (\C^{n+1} \setminus \{0\})/\C^*,
\end{equation}
where $\C^*$ acts via homotheties, i.e., $\lambda \cdot
(a_0,\dots,a_n) = (\lambda a_0,\dots, \lambda a_n)$ for $\lambda \in
\C^*$ and $(a_0,\dots,a_n) \in \C^{n+1}$.  Thus $(a_0,\dots,a_n)$ are
\emph{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 $\P^n$.

\subsection*{\emph{Homogeneous Coordinate Rings}} A projective variety
$V \subseteq \P^{\,n}$ is defined by the vanishing of finitely many
homogeneous polynomials in the polynomial ring $S =
\C[x_0,\dots,x_n]$.  The \emph{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 $V$.

\bigbreak\goodbreak

The polynomial ring $S$ is graded by setting $\deg(x_i) = 1$.  This
gives the decomposition $S = \bigoplus_{d=0}^\infty S_d$, 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 $\V(I) \subseteq S =\C[x_0,\dots,x_n]$ also defines an
affine variety $\widehat{V} \subseteq \C^{n+1}$, called the
\emph{affine cone} of $V$.  The variety $\widehat{V}$ satisfies
\begin{equation}
\label{1.vconev}
V = (\widehat{V} \setminus \{0\})/\C^*,
\end{equation}
and its coordinate ring is the homogeneous coordinate ring of $V$,
i.e.,
\[
\C[\widehat{V}] = \C[V].
\]

\begin{example}
\label{2.rnccurve}
In Example~\ref{1.rnccone} of Chapter~\ref{affine} we encountered
ideal ideal
\[
I = \langle x_i x_{j+1} - x_{i+1} x_j \mid 0 \le i < j \le
d-1\rangle \subseteq  \C[x_0,\dots,x_d]
\]
generated by the 
$2\times 2$
minors of the matrix
\[
\begin{pmatrix} x_0 & x_1 & \cdots & x_{d-2} & x_{d-1} \\
x_1 & x_2 & \cdots & x_{d-1} & x_{d}
\end{pmatrix}.
\]
Since $I$ is homogeneous, it defines a projective variety $C_d
\subseteq \P^d$ that is the image of the map
\[
\Phi: \P^1 \longrightarrow \P^d
\]
defined in homogeneous coordinates by $(s,t) \mapsto
(s^d,s^{d-1}t,\dots,st^{d-1},t^d)$ (see Exercise~\ref{1.1.1} of
Chapter~\ref{affine}).  This shows that $C_d$ is a curve, called the
\emph{rational normal curve} of degree $d$.  Furthermore, the affine
cone of $C_d$ is the rational normal cone $\widehat{C}_d \subseteq
\C^{d+1}$ discussed in Example~\ref{1.rnccone} of
Chapter~\ref{affine}.

We know from Chapter~\ref{affine} that $\widehat{C}_d$ is an affine
toric surface; we will soon see that $C_d$ is a projective toric
curve. \eoe
\end{example}

\begin{example}
\label{2.xyzw}
The affine toric variety $\V(xy-zw) \subseteq \C^4$ studied in
Chapter~\ref{affine} is the affine cone of the rojective surface $V =
\V(xy-zw) \subseteq \P^3$.  Recall that this surface is isomorphic to
$\P^1\times\P^1$ via the Segre embedding
\[
\P^1\times\P^1 \longrightarrow \P^3
\]
given by $(s,t;u,v) \mapsto (su,tv,sv,tu)$.  We will see below that $V
\simeq \P^1\times\P^1$ is the projective toric variety coming from 
the unit square in the plane. \eoe
\end{example}

As in the affine case, a projective variety $V \subseteq \P^{\,n}$ has the
\emph{classical topology}, induced from the usual topology on $\P^{\,n}$,
and the \emph{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.  

\subsection*{\emph{Rational Functions on Irreducible Projective
Varieties}} A homogeneous polynomial $f \in S$ of degree $d$ does not
give a function on $\P^n$ since
\[
f(\lambda x_0,\dots,\lambda x_n) = \lambda^d f(x_0,\dots,x_n).
\]
However, the quotient of two such polynomials $f,g \in S_d$ gives the
well-defined function
\[
\frac{f}{g} : \P^n \setminus \V(g) \to \C.
\]
provided $g \ne 0$.  We write this as $f/g : \P^n \dashrightarrow \C$
and say that $f/g$ is a \emph{rational function} on $\P^n$.

More generally, suppose that $V \subseteq \P^n$ is irreducible, and let
$f,g \in \C[V] = \C[\widehat{V}]$ be homogeneous of the same degree
with $g \ne 0$.  Then $f$ and $g$ give functions on the affine cone
$\widehat{V}$ and hence an element $f/g \in \C(\widehat{V})$.  By
\eqref{1.vconev}, this induces a rational function $f/g : V
\dashrightarrow \C$.  Thus
\[
\C(V) = \{f/g \in \C(\widehat{V}) \mid f,g \in
\C[V] \text{ homogeneous of the same degree, } g \ne 0\}
\]
is the field of rational functions on $V$.  It is customary to write
the set on the left as $\C(\widehat{V})_0$ since it consists of the
degree $0$ elements of $\C(\widehat{V})$.

\subsection*{\emph{Affine Pieces of Projective Varieties}} A
projective variety $V \subseteq \P^n$ is a union of Zariski open sets
that are affine.  To see why, let $U_i = \P^{\,n} \setminus \V(x_i)$.
Then $U_i \simeq \C^n$ via the map
\begin{equation}
\label{2.uimap}
(a_0,\dots,a_n) \longmapsto
(a_0/a_i,\dots,a_{i-1}/a_i,a_{i+1}/a_i,\dots,a_n/a_i)
\end{equation}
so that in the notation of Chapter~\ref{affine}, we have
\[
U_i = \Spec(\C[x_0/x_i,\dots,x_{i-1}/x_i,x_{i+1}/x_i,\dots,x_n/x_i]).
\]
Then $V\cap U_i$ is a Zariski open subset of $V$ that maps via
\eqref{2.uimap} to the affine variety in $\C^n$ defined by the
equations
\begin{equation}
\label{2.definevui}
f(x_0/x_i,\dots,x_{i-1}/x_i,1,x_{i+1}/x_i,\dots,x_n/x_i) = 0
\end{equation}
as $f$ varies over all homogeneous polynomials in $\I(V)$.  

We call $V\cap U_i$ an \emph{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 $\bar{x}_i \in
\C[V] = \C[\widehat{V}] \subseteq \C(\widehat{V})$, so that we get the
localization
\begin{equation}
\label{2.localize}
\C[V]_{\bar{x}_i} = \{f/\bar{x}_i^{\,k} \mid f \in \C[V],\ k \ge 0\}
\subseteq \C(\widehat{V})
\end{equation}
as in equation \eqref{1.localize} of Chapter~\ref{affine}.  Then
\begin{equation}
\label{2.affinepiece}
(\C[V]_{\bar{x}_i})_0 = \{f/\bar{x}_i^{\,k} \in \C[V]_{\bar{x}_i} \mid
f \text{ is homogeneous of degree } k\}
\end{equation}
is the subring of $\C[V]_{\bar{x}_i}$ consisting of all element of
degree~$0$.  Note that \eqref{2.affinepiece} uses only the homogeneous
coordinate ring $\C[V]$ and the element $\bar{x}_i$.  Yet it gives an
affine piece of $V$ as follows.

\begin{lemma}
\label{2.coordringvui}
Assume that $V$ is irreducible and define $U_i$ and
$(\C[V]_{\bar{x}_i})_0$ as above.  Then the affine piece $V\cap U_i$
of $V$ has coordinate ring
\[
\C[V\cap U_i] \simeq (\C[V]_{\bar{x}_i})_0.
\]
\end{lemma}

\begin{proof}
We have an exact sequence
\[
0 \longrightarrow \I(V) \longrightarrow \C[x_0,\dots,x_n]
\longrightarrow \C[V] \longrightarrow 0.
\]
If we localize at $x_i$ as in \eqref{2.localize}, we get
\[
0 \longrightarrow \I(V)_{x_i} \longrightarrow
\C[x_0,\dots,x_n]_{x_i} \longrightarrow \C[V]_{\bar{x}_i} \longrightarrow 0
\]
which is exact since localization preserves exactness (see
Exercise~\ref{2.0.2}).  Taking elements of degree 0 gives the exact
sequence
\[
0 \longrightarrow (\I(V)_{x_i})_0 \longrightarrow
(\C[x_0,\dots,x_n]_{x_i})_0 \longrightarrow (\C[V]_{\bar{x}_i})_0
\longrightarrow 0.
\]
%Note $(\C[x_0,\dots,x_n]_{x_i})_0 =
%\C[x_0/x_i,\dots,x_{i-1}/x_i,x_{i+1}/x_i,\dots,x_n/x_i])$.  
%Since elements of $(\I(V)_{x_i})_0$ are of the form
%$f/x_i^{\,k}$ where $f \in \I(V)$ is homogeneous of degree
%$k$, we have
Note that $(\C[x_0,\dots,x_n]_{x_i})_0 =
\C[x_0/x_i,\dots,x_{i-1}/x_i,x_{i+1}/x_i,\dots,x_n/x_i])$.  If $f \in
\I(V)$ is homogeneous of degree $k$, then
\[
f/x_i^{\,k} = f(x_0/x_i,\dots,x_{i-1}/x_i,1,x_{i+1}/x_i,\dots,x_n/x_i)
\in (\I(V)_{x_i})_0.
\]
Thus $(\I(V)_{x_i})_0$ comes from the defining equations
\eqref{2.definevui} of $V\cap U_i$.  Since $\C[V]$ is an integral
domain, the same is true for $(\C[V]_{\bar{x}_i})_0$.  It follows that
$(\I(V)_{x_i})_0$ is a prime ideal, which by the Nullstellensatz
implies that $(\I(V)_{x_i})_0 = \I(V\cap U_i)$.  The lemma follows
immediately.
\end{proof}

One can also explore what happens when we intersect affine pieces $V
\cap U_i$ and $V \cap U_i$ for $i \ne j$.  By Exercise~\ref{2.0.3}, $V
\cap U_i \cap U_j$ is affine with coordinate ring
\begin{equation}
\label{2.vuiuj}
\C[V \cap U_i \cap U_j] \simeq (\C[V]_{x_ix_j})_0
\end{equation}
when $V$ is irreducible. Actually, irreducibility is not essential
here---\eqref{2.vuiuj} and Lemma~\ref{2.coordringvui} hold for
arbitrary varieties, provided one uses the more general notion of
localization mentioned in Exercise~\ref{1.0.2} of
Chapter~\ref{affine}.

We will use these results in \S2 when we explore the structure of
projective toric varieties.  We will also see later in the book that
Lemma~\ref{2.coordringvui} 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.

\subsection*{\emph{Products of Projective Spaces}} One can study 
the product $\P^n\times\P^m$ of projective spaces using the
bigraded ring $\C[x_0,\dots,x_m,y_0,\dots,y_m]$, 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 $\P^n\times\P^m$.  This allows us to define
varieties in $\P^n\times\P^m$ using bihomogeneous ideals.  In
particular, the ideal $\I(V)$ of a variety $V \subseteq
\P^n\times\P^m$ is a bihomogeneous ideal.

Another way to study $\P^n\times\P^m$ is via the \emph{Segre
embedding} 
\[
\P^n\times\P^m \longrightarrow \P^{nm + n + m}
\]
defined by mapping $(a_0,\dots,a_n,b_0,\dots,b_m)$ to the point
\[(a_0b_0,a_0b_1,\dots,a_0b_m,a_1b_0, \dots,a_1b_m,a_nb_m,a_1b_0,
\dots,a_nb_m).
\]
This map is studied in \cite[Exercise~14 to Chapter~8, \S4]{iva}.  If
$\P^{nm + n + m}$ has homogeneous coordinates $x_{ij}$ for $0 \le i
\le n, 0 \le j \le m$, then $\P^n\times\P^m \subseteq
\P^{nm + n + m}$ is define by vanishing of the $2\times2$ minors of the
$(n+1)\times(m+1)$ matrix of variables
\[
\begin{pmatrix} x_{00} & \cdots & x_{0m} \\ \vdots && \vdots \\
x_{n0} & \cdots & x_{nm}\end{pmatrix}.
\]
This follows since an $(n+1)\times(m+1)$ 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 $\P^n\times\P^m$.  A homogeneous polynomial $F(x_{ij})$
of degree $d$ gives the bihomogeneous polynomial $F(x_iy_j)$ of
bidegree $(d,d)$.  Hence any subvariety of $\P^{nm + n + m}$ lying in
$\P^n\times\P^m$ can be defined by a bihomogeneous ideal in
$\C[x_0,\dots,x_m,y_0,\dots,y_m]$.  Going the other way takes
more thought and is discussed in Exercise~\ref{2.0.segre}.

We also have the following useful result proved in
Exercise~\ref{2.0.prodsubexer}.

\begin{proposition}
\label{2.prodsub}
Given subvarieties $V \subseteq \P^n$ and $W \subseteq \P^m$, the
product $V\times W$ is a subvariety of $\P^n\times\P^m$.\qed
\end{proposition}

\subsection*{\emph{Weighted Projective Space}} The graded ring
associated to $\P^n$ is $\C[x_0,\dots,x_n]$, where each variable $x_1$
has degree $1$.  More generally, let $q_0,\dots,q_n$ be positive
integers with $\gcd(q_0,\dots,q_n) = 1$ and define
\[
\P(q_0,\dots,q_n) = (\C^{n+1} \setminus \{0\})/\sim
\]
where $\sim$ is the equivalence relation
\[
(a_0,\dots,a_n) \sim (b_0,\dots,b_n) \iff a_i = \lambda^{q_i} b_i,\ i
= 0,\dots,n \text{ for some } \lambda \in \Cstar.
\]
We call $\P(q_0,\dots,q_n)$ a \emph{weighted projective space}.  Note
that $\P^n = \P(1,\dots,1)$.   

The ring corresponding to $\P(q_0,\dots,q_n)$ is the
graded ring $\C[x_0,\dots,x_n]$ where $x_i$ now has degree $q_i$.  A
polynomial $f$ is \emph{weighted homogeneous} of degree $d$ if every
monomial $x^\a$ appearing in $f$ satisfies $\a\cdot(q_0,\dots,q_n) =
d$.  The $f = 0$ is well-defined on $\P(q_0,\dots,q_n)$ when $f$ is
\emph{weighted homogeneous}, so that one can define varieties in
$\P(q_0,\dots,q_n)$ using weighted homogeneous ideals of
$\C[x_0,\dots,x_n]$.  

\begin{example}
\label{2.p112}
We can embed the weighted projective plane $\P(1,1,2)$ in $\P^3$ using
the monomials $x_0^2,x_0x_1,x_1^2,x_2$ of weighted degree $2$.  In
other words, the map
\[
\P(1,1,2) \longrightarrow \P^3
\]
given by
\[
(a_0,a_1,a_2) \longmapsto (a_0^2,a_0a_1,a_1^2,a_2)
\]
is well-defined and injective.  One can check that this map induces
\[
\P(1,1,2) \simeq \V(y_0y_2 - y_1^2) \subseteq \P^3,
\]
where $y_0,y_1,y_2,y_3$ are homogeneous coordinates on $\P^3$.\eoe
\end{example}

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

\exercises

\small

\begin{exercise}
\label{2.0.2}
This exercise is concerned with localization over an integral domain
$R$.  
\begin{thmenum}
\item Let $f \in R \setminus \{0\}$ and let $0 \to M_1 \to M_2 \to
  M_3$ be an exact sequence of $R$-modules.  Prove that
\[
0 \longrightarrow M_1\otimes_R R_f \longrightarrow M_2\otimes_R R_f
\longrightarrow M_3\otimes_R R_f\longrightarrow 0
\]
is exact, where $R_f$ is the localization of $R$ at $f$ defined in
\eqref{1.localize} of Chapter~\ref{affine}.
\item Now let $R = \C[x_0,\dots,x_n]$ and let $\C[V] = R/\I(V)$ for an
irreducible projective variety $V \subseteq \P^n$.  Also let $\bar{x}_i$
be the image of $x_i$ in $\C[V]$.  Note the $\C[V]$ is an $R$-module.
Prove that $\C[V]_{\bar{x}_i} \simeq \C[V] \otimes_R R_{x_i}$.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{2.0.3}
Prove the claim made in \eqref{2.vuiuj}.
\end{exercise}

\begin{exercise}
\label{2.0.4}
Let $V \subseteq \P^n$ be a projective variety.  Take $f_0,\dots,f_m
\in S_d$ such that the intersection $V\cap\V(f_0,\dots,f_m)$ is empty.
Prove that the map
\[
(a_0,\dots,a_n) \longmapsto
(f_0(a_0,\dots,a_n),\dots,f_m(a_0,\dots,a_n))
\]
induces a well-defined map function $\Phi : V \to \P^m$.  
\end{exercise}

\begin{exercise}
\label{2.0.segre}
Let $V \subseteq \P^n\times\P^m$ be defined by $f_\ell(x_i,y_j) = 0$,
where $f_\ell(x_i,y_j)$ is bihomogenous of bidegree $(a_\ell,b_\ell)$,
$\ell = 1,\dots,s$.  The goal of this exercise is to show that when we
embed $\P^n\times\P^m$ in $\P^{nm+n+m}$ via the Segre embedding
described in the text, $V$ becomes a subvariety of $\P^{nm+n+m}$.
\begin{thmenum}
\item For each $\ell$, pick an integer $d_\ell \ge
\max\{a_\ell,b_\ell\}$ and consider the polynomials $f_{\ell,\a,\b} =
x^\a y^\b f_i(x_i,y_j)$ where $\ell = 0,\dots,s$ and $|\a| = d_\ell -
a_\ell, \ |\b| = d_\ell - b_\ell$.  Note that $f_{\ell,\a,\b}$ is
bihomogenous of bidegree $(d_\ell,d_\ell)$.  Prove that $V \subseteq
\P^n\times\P^m$ is defined by the vanishing of the $f_{\ell,\a,\b}$.
\item Use part (a) to show that $V$ is a subvariety of
  $\P^{nm+n+m}$ under the Segre embedding.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{2.0.prodsubexer}
Prove Proposition~\ref{2.prodsub}
\end{exercise}

\begin{exercise}
\label{2.0.segrep1p1}
Consider the Segre embedding $\P^1\times\P^1 \to \P^3$.  Show that after
relabeling coordinates, the affine cone of $\P^1\times\P^1$ in
$\P^3$ is the variety $\V(xy-zw) \subseteq \C^4$ featured in many
examples in Chapter~\ref{affine}.
\end{exercise}

\normalsize

\section{Lattice Points and Projective Toric Varieties}

We first observe that $\P^n$ is a toric variety with torus
\begin{align*}
T_{\P^n} &= \P^n \setminus \V(x_0\dots x_n) = \{(a_0,\dots,a_n) \in
\P^n \mid a_0\dots a_n \ne 0\}\\ &= \{(1,t_1,\dots,t_n) \in \P^n \mid
t_1,\dots,t_n \in \Cstar\} \simeq \torus.
\end{align*}
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
\[
1 \longrightarrow \Cstar \longrightarrow (\Cstar)^{n+1}
\stackrel{\pi}{\longrightarrow} T_{\P^n} \longrightarrow 1
\]
coming from the definition \eqref{2.pndef} of $\P^n$.  Hence the
character lattice of $T_{\P^n}$ is
\begin{equation}
\label{2.mndef}
\cM_n = \{(a_0,\dots,a_n) \in \Z^{n+1} \mid {\textstyle \sum_{i=0}^n}
a_i = 0\},
\end{equation}
and the lattice of one-parameter subgroups $\cN_n$ is the quotient
\[
\cN_n = \Z^{n+1}/\Z(1,\dots,1).
\]

\subsection*{\emph{Lattice Points and Projective Toric Varieties}}
Let $T_N$ be a torus with lattices $M$ and $N$ as usual.  In
Chapter~\ref{affine}, we used a finite set of lattice points of $\cA =
\{m_1,\dots,m_s\} \subseteq M$ to create the affine toric variety
$\ya$ as the Zariski closure of the image of the map
\[
\Phi_\cA : T_N \longrightarrow \C^{s},\quad t \longmapsto
(\chi^{m_1}(t),\dots,\chi^{m_s}(t)).
\]
To get a projective toric variety, we regard $\Phi_\cA$ as a map to
$(\Cstar)^s$ and compose with the homomorphism $\pi : (\Cstar)^s \to
T_{\P^{s-1}}$ to obtain
\begin{equation}
\label{2.latticemap}
T_N \stackrel{\Phi_\cA}{\longrightarrow} \C^{s}
\stackrel{\pi}{\longrightarrow}T_{\P^{s-1}} \subseteq \P^{s-1}.
\end{equation}

\begin{definition}
\label{2.latticeptdef}
Given a finite set $\cA \subseteq M$, the \emph{\bfseries projective
toric variety} $\xa$ is defined to be the Zariski closure in $\P^{s-1}$
of the image of the map $\Phi_\cA$ from \eqref{2.latticemap}.
\end{definition}

\begin{proposition}
$\xa$ is a toric variety of dimension equal to the dimension of the
  smallest affine subspace of $\mr$ containing $\cA$.
\end{proposition}

\begin{proof}
The proof that $\xa \subseteq \P^{s-1}$ is a toric variety is
similar to the proof given in Propostion~\ref{1.lattpt} of
Chapter~\ref{affine} that $\ya \subseteq \P^{s}$ is a toric variety.
The assertion concerning the dimension of $\xa$ will follow from
Proposition~\ref{2.torusxa} below.
\end{proof}

More concretely, $\xa$ is the Zariski closure of the image of the map
\[
T_N \longrightarrow \P^{s-1},\quad t \longmapsto
(\chi^{m_1}(t),\dots,\chi^{m_s}(t))
\]
given by the characters coming from $\cA = \{m_1,\dots,m_s\} \subseteq
M$.  In particular, if $M = \Z^n$, the $\chi^{m_i}$ is the Laurent
monomial $t^{m_i}$ and $\xa$ as the Zariski closure of the image of
\[
T_N \longrightarrow \P^{s-1},\quad t \longmapsto (t^{m_1},\dots,t^{m_s}).
\]
In the literature, $\cA \subseteq \Z^n$ is often given as an
$n\times s$ integer matrix, so that the elements of $\cA$ are the
columns of the matrix.

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

\bigbreak\goodbreak

\begin{example}
\label{2.perm1}
Let $M = \Z^{3\times3}$ be the lattice of $3\times3$ integer matrices
and let
\[
\cP_3 = \{ 3\times3 \text{ permutation matrices}\} \subseteq
\Z^{3\times3}. 
\]
Write $\C[M]) = \C[t_1^{\pm1},\dots,t_9^{\pm1}]$, where the variables
give the generic $3\times3$ matrix
\[
\begin{pmatrix}
t_1 & t_2 & t_3\\ t_4 & t_5 & t_6\\ t_7 & t_8 & t_9
\end{pmatrix}
\]
with nonzero entries.  Also let $\P^5$ have homogeneous coordinates
$x_{ijk}$ indexed by triples such that $\big(\begin{smallmatrix} 1 & 2
& 3\\ i & j & k\end{smallmatrix}\big)$ is a permutation in $S_3$.
Then $X_{\cP_3} \subseteq \P^5$ is the Zariski closure of the image of
the map $T_N \to \P^5$ given by the Laurent monomials
$t_it_jt_k$ for $\big(\begin{smallmatrix} 1 & 2 & 3\\ i & j &
k\end{smallmatrix}\big) \in S_3$.  The ideal of $X_{\cP_3}$ is
\[
\I(X_{\cP_3}) = \langle x_{123}x_{231}x_{312} -
x_{132}x_{321}x_{213}\rangle \subseteq \C[x_{ijk}],
\]
where the relation comes from the fact that three of the permutation
matrices sum to the other three (Exercise~\ref{2.1.1}).  Ideals of the
toric varieties arising from permutation matrices have applications to
sampling problems in statistics \cite[p.\ 148]{sturmfels}.\eoe
\end{example}

\subsection*{\emph{The Affine Cone of a Projective Toric Variety}}
The projective variety $\xa \subseteq \P^{s-1}$ has an affine cone
$\widehat{X}_\cA \subseteq \C^{s}$.  How does $\widehat{X}_\cA$
relate to the affine toric variety $\ya \subseteq \C^{s}$ constructed
in Chapter~\ref{affine}? 

Recall from Chapter~\ref{affine} that when $\cA = \{m_1,\dots,m_s\}
\subseteq M$, the map $e_i \mapsto m_i$ induces an exact sequence
\begin{equation}
\label{2.Ldef}
0 \longrightarrow L \longrightarrow \Z^s \longrightarrow M
\end{equation} 
and that the ideal of $\ya$ is the toric ideal
\[
I_L = \big\langle x^{\alpha} - x^{\beta} \mid \alpha,\beta \in \N^s\
\text{\rm and } \alpha -\beta \in L\big\rangle
\]
(Proposition~\ref{1.toricid} of Chapter~\ref{affine}).  Then we have
the following result.

\bigbreak\goodbreak

\begin{proposition}
\label{2.whenhomo}
Given $\ya$, $\xa$ and $I_L$ as above, the following are equivalent:
\begin{thmenum}
\item $\ya \subseteq \C^s$ is the affine cone $\widehat{X}_\cA$ of
  $\xa \subseteq \P^{s-1}$.
\item $I_L = \I(\xa)$.
\item $I_L$ is homogeneous.
\item There is $u \in N$ and $k > 0$ in $\N$ such that $\langle
  m_i,u\rangle = k$ for $i = 1,\dots,s$.
\end{thmenum}
\end{proposition}

\begin{proof}
The equivalence (a) $\Leftrightarrow$ (b) follows from the
equalities $\I(\xa) = \I(\widehat{X}_\cA)$ and $I_L = \I(\ya)$, and
the implication (b) $\Rightarrow$ (c) is obvious.  

For (c) $\Rightarrow$ (d), assume that $I_L$ is homogeneous and take
$x^\a - x^\b \in L$ for $\a - \b \in L$.  If $x^\a$ and $x^\b$ had
different degrees, then $x^\a, x^\b \in I_L = \I(\ya)$ would vanish on
$\ya$.  This is impossible since $(1,\dots,1) \in \ya$ by
\eqref{2.latticemap}.  Hence $x^\a$ and $x^\b$ have the same degree,
which implies that $\ell\cdot(1,\dots,1) = 0$ for all $\ell \in L$.
Now tensor \eqref{2.Ldef} with $\Q$ and take duals to obtain an exact
sequence
\[
N_\Q \longrightarrow \Q^s \longrightarrow \mathrm{Hom}_\Q(L_\Q,\Q)
\longrightarrow 0.
\]
The above argument shows that $(1,\dots,1) \in \Q^s$ maps to zero in
$\mathrm{Hom}_\Q(L_\Q,\Q)$ and hence comes from an element $\tilde{u}
\in N_\Q$.  In other words, $\langle m_i,\tilde{u}\rangle = 1$ for all
$i$.  Clearing denominators gives the desired $u \in N$ and $k > 0$ in
$\N$.

Finally, we prove (d) $\Rightarrow$ (b).  Since $I_L = \I(\ya)$, it
suffices to show that
\[
\widehat{X}_\cA \cap (\Cstar)^s \subseteq \ya.  
\]
Let $p \in \widehat{X}_\cA \cap (\Cstar)^s$.  Since $\xa \cap
T_{\P^{s-1}}$ is the torus of $\xa$, it follows that
\[
p = \mu\cdot(\chi^{m_1}(t),\dots,\chi^{m_s}(t))
\]
for some $\mu \in \Cstar$ and $t \in T_N$.  Let $u \in N$ be as in
part (d).  This gives a one-parameter subgroup of $T_N$, which we
write as $\tau \mapsto \lambda^u(\tau)$ for $\tau \in \Cstar$.  Then
$\lambda^u(\tau) t \in T_N$ maps to the point $q \in \ya$ given by
\[
q = \big(\chi^{m_1}(\lambda^u(\tau)
t),\dots,\chi^{m_s}(\lambda^u(\tau) t)\big) = \big(\tau^{\langle
m_1,u\rangle} \chi^{m_1}(t),\dots,\tau^{\langle m_s,u\rangle}
\chi^{m_s}(t)\big),
\]
since $\chi^m(\lambda^u(\tau)) = \tau^{\langle m,u\rangle}$ by the
description of $\langle \ ,\ \rangle$ given in \S1 of
Chapter~\ref{affine}.  The hypothesis of part (d) allows us to rewrite
$q$ as
\[
q = \tau^k\cdot(\chi^{m_1}(t),\dots,\chi^{m_s}(t)).
\]
Using $k > 0$, we can choose $\tau$ so that $p = q \in \ya$.  This
completes the proof.
\end{proof}

The condition $\langle m_i,u\rangle = k$, $i = 1,\dots,s$ for some $u
\in N$ and $k > 0$ in $\N$ means that $\cA$ lies in an affine
hyperplane of $M_\Q$ not containing the origin.  When $M = \Z^n$ and
$\cA$ consists of the columns of an $n\times s$ integer matrix, this
condition is equivalent to $(1,\dots,1)$ lying in the row space of the
matrix (Exercise~\ref{2.1.2}).

\begin{example}
\label{2.poly.rnc}
We will examine the rational normal curve $C_d \subseteq \P^d$ using two
different sets of lattice points.  

First let $\cA$ consist of the columns of the $2\times (d+1)$ matrix
\[
A = \begin{pmatrix}
d & d-1 & \cdots & 1 & 0\\ 0 & 1 & \cdots & d-1 & d.
\end{pmatrix}
\]
The columns give the Laurent monomials defining the rational normal
curve $C_d$ in Example~\ref{2.rnccurve}.  It follows that $C_d$ is a
projective toric variety.  The ideal of $C_d$ is the homogeneous ideal
given in Example~\ref{2.rnccurve}, and the corresponding affine
hyperplane of $\Z^2$ containing $\cA$ (= the columns of $A$) consists
of all points $(a,b)$ satisfying $a+b = d$.  It is equally easy to see
that $(1,\dots,1)$ is in the row space of $A$.  In particular, we have
\[
\xa = C_d\quad\text{and}\quad  \ya = \widehat{C}_d.
\]

Now let $\cB = \{0,1,\dots,d-1,d\} \subseteq \Z$.  This gives the map
\[
\Phi_\cB : \Cstar \longrightarrow \P^d, \quad t \mapsto
(1,t,\dots,t^{d-1},t^d).
\]
The resulting projective variety is the rational normal curve, i.e.,
$X_\cB = C_d$, but the affine variety of $\cB$ is \emph{not} the
rational normal cone, i.e., $Y_\cB \ne \widehat{C}_d$.  This can be
seen by observing that $x_1^2-x_2$ vanishes at
$(1,t,\dots,t^{d-1},t^d) \in \C^{d+1}$ for all $t \in \Cstar$.  Thus
$x_1^2-x_2 \in \I(Y_\cB)$, so that $\I(Y_\cB)$ is not homogeneous.\eoe
\end{example} 

Given any $\cA \subseteq M$, there is a standard way to modify
$\cA$ so that the conditions of Proposition~\ref{2.whenhomo} are met:
use $\cA\times\{1\} \subseteq M\oplus\Z$.  This lattice corresponds
to the torus $T_n\times\Cstar$, and since
\begin{equation}
\label{2.ax1}
\Phi_{\cA\times\{1\}}(t,\mu) =
(\chi^{m_1}(t)\mu,\dots,\chi^{m_s}(t)\mu) =
\mu\cdot(\chi^{m_1}(t),\dots,\chi^{m_1}(t)),
\end{equation}
it follows immediately that $X_{\cA\times\{1\}} = \xa \subseteq
\P^{s-1}$.  Since $\cA\times\{1\}$ lies in an affine hyperplane
missing the origin, Proposition~\ref{2.whenhomo} implies that $\xa$ has
affine cone $Y_{\cA\times\{1\}} = \widehat{X}_\cA$.  When $M = \Z^n$
and $\cA$ is represented by the columns of an $n\times s$ integer
matrix, we obtain $\cA\times\{1\}$ by adding a row of $1$s to the
matrix. 

\subsection*{\emph{The Torus of a Projective Toric Variety}} Our next
task is to determine the torus of $\xa$.  We will do so by identifying
the character lattice.  This will also tell us the dimension of $\xa$.
Given $\cA = \{m_1,\dots,m_s\} \subseteq M$, we set
\[
\Z'\!\cA = \big\{{\textstyle \sum_{i=1}^s} a_i m_i \mid a_i \in \Z,\
{\textstyle \sum_{i=1}^s} a_i = 0\big\}.
\]
The rank of $\Z'\!\cA$ is the dimension of the smallest affine subspace
of $\mr$ containing the set $\cA$ (Exercise~\ref{2.1.dimzp}).

\begin{proposition}
\label{2.torusxa} 
Let $\xa$ be the projective toric variety of $\cA \subseteq M$.  Then:
\begin{thmenum}
\item The lattice $\Z'\!\cA$ is the character lattice of torus
of $\xa$.
\item The dimension of $\xa$ is the dimension of the smallest affine
subspace of $\mr$ containing $\cA$.  In particular,
\[
\dim \xa = \begin{cases} \mathrm{rank}\,\Z\cA - 1 & \text{if } \ya
  \text{ is the affine cone of } \xa\\ \mathrm{rank}\,\Z\cA &
  \text{otherwise.} \end{cases}
\]
\end{thmenum}
\end{proposition}

\begin{proof} 
To prove part (a), let $M(T_{\xa})$ be the character lattice of the
torus $T_{\xa}$ of $\xa$.  By \eqref{2.latticemap}, we have the
commutative diagram
\[
\SelectTips {cm}{}
\xymatrix{
T_N \ar[r] \ar@{->>}[rd] & \P^{s-1} \\
& T_{\xa}. \ar@{^{(}->}[u]
}
\]
which induces the commutative diagram of character lattices
\[
\SelectTips {cm}{}
\xymatrix{
M  & \cM_{s-1} \ar[l] \ar@{->>}[d] \\
& M(T_\xa) \ar@{_{(}->}[lu]
}
\]
since $\cM_{s-1} = \{(a_1,\dots,a_s) \in \Z^{s} \mid \sum_{i=0}^s a_i
m_i = 0\}$ is the character lattice of $T_{\P^{s-1}}$ by
\eqref{2.mndef}.  The map $\cM_{s-1} \to M$ is induced by the map
$\Z^s \to M$ that sends $e_i$ to $m_i$.  Thus $\Z'\!\cA$ is the image of
$\cM_{s-1} \to M$ and hence equals $M(T_{\xa})$ by the above diagram.

The first assertion of part (b) follows from part (a) and the
observation that $\mathrm{rank}\,\Z'\!\cA$ is the dimension of the
smallest affine subpace of $\mr$ containing $\cA$.  Furthermore, if
$\ya$ equals the affine cone of $\xa$, then there is $u \in N$ with
$\langle m_i,u\rangle = k > 0$ for all $i$ by
Proposition~\ref{2.whenhomo}.  Thus $\langle \sum_{i=1}^s a_i
m_i,u\rangle = k(\sum_{i=1}^s a_i)$, which gives the exact sequence
\[
0 \longrightarrow \Z'\!\cA \longrightarrow \Z\cA \stackrel{\langle\
,u\rangle}{\longrightarrow} k\Z \longrightarrow 0.
\]
Then $k>0$ implies $\mathrm{rank}\,\Z\cA - 1 = \mathrm{rank}\,\Z'\!\cA =
\dim\xa$.  On the other hand, if $\ya \ne \widehat{X}_\cA$, then the
ideal $I_L$ is not homogeneous, where $L$ is defined by
\[
0 \longrightarrow L \longrightarrow \Z^s \longrightarrow \Z\cA
\longrightarrow 0.
\]
by \eqref{2.Ldef}.  Using the usual description of $I_L$, it follows
that $m\cdot(1,\dots,1) \ne 0$ for some $m \in L$.  Hence there is
$\ell > 0$ such that we have an diagram 
\[
\SelectTips {cm}{}
\xymatrix@C=8pt@R=8pt{
 & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] & \\
0 \ar[r] & L\cap\cM_{s-1} \ar[r] \ar[d] & L \ar[r] \ar[d] & \ell \Z
\ar[r] \ar[d] & 0\\
0 \ar[r] & \cM_{s-1} \ar[r] \ar[d] & \Z^s \ar[r] \ar[d] & \Z
\ar[r] \ar[d] & 0\\
0 \ar[r] & \Z'\!\cA \ar[r] \ar[d] & \Z\cA \ar[r] \ar[d] & \Z/\ell\Z
\ar[r] \ar[d] & 0\\
 & 0  & 0  & 0  &
}
\]
with exact rows and columns.  Hence $\mathrm{rank}\,\Z\cA =
\mathrm{rank}\,\Z'\!\cA = \dim\xa$.
\end{proof}

\begin{example}
\label{2.zprimeaex}
Let $\cA = \{e_1,e_2,e_1+2e_2,2e_1+e_2\} \subseteq \Z^2$.  One
computes that $\Z\cA = \Z^2$ but $\Z'\!\cA = \{(a,b) \in \Z^2 \mid
a+b\equiv 0 \bmod 2\}$.  Thus $\Z'\!\cA$ has index~$2$ in $\Z\cA$.
This means that $\ya \ne \widehat{X}_\cA$ and the map of tori
\[
T_{\ya} \longrightarrow T_{\xa}
\]
is two-to-one, i.e., its kernel has order $2$
(Exercise~\ref{2.1.zprimeaexer}). \eoe
\end{example}

\subsection*{\emph{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 $\xa
\subseteq \P^{s-1}$.

The affine open set $U_i = \P^{s-1} \setminus \V(x_i)$ contains the
torus $T_{\P^{s-1}}$.  Thus
\[
T_{\xa} = \xa \cap T_{\P^{s-1}} \subseteq \xa \cap U_i.  
\]
Since $\xa$ is the Zariski closure of $T_{\xa}$ in $\P^{s-1}$, it
follows that $\xa \cap U_i$ is the Zariski closure of $T_{\xa}$ in
$U_i \simeq \C^{s-1}$.  Thus $\xa \cap U_i$ is an affine toric variety.

Given $\cA = \{m_1,\dots,m_s\} \subseteq \mr$, the affine semigroup
associated to $\xa \cap U_i$ is easy to determine.  Recall that $U_i
\simeq \C^{s-1}$ is given by
\[(a_1,\dots,a_s) \longmapsto
(a_1/a_i,\dots,a_{i-1}/a_i,a_{i+1}/a_i,\dots,a_s/a_i).
\]
Combining this and $\chi^{m_j}/\chi^{m_i} = \chi^{m_j-m_i}$ with the
map \eqref{2.latticemap}, we see that $\xa \cap U_i$ is the Zariski
closure of the image of the map
\[
T_N \longrightarrow \C^{s-1}
\]
given by
\begin{equation}
\label{2.txaui}
t \longmapsto \big(\chi^{m_1-m_i}(t),\dots,\chi^{m_{i-1}-m_i}(t),
\chi^{m_{i+1}-m_i}(t), \dots, \chi^{m_s-m_i}(t)\big).
\end{equation}
If we set $\cA_i = \cA - m_i = \{m_j - m_i \mid j \ne i\}$, it
follows that
\[
\xa \cap U_i = X_{\cA_i} = \Spec(\C[\sS_i]),
\]
where $\sS_i = \N\cA_i$ is the affine semigroup generated by $\cA_i$.
We have thus proved the following result.

\begin{proposition}
\label{2.localpieces}
Let $\xa \subseteq \P^{s-1}$ for $\cA = \{m_1,\dots,m_s\} \subseteq
\mr$.  Then the affine piece $\xa\cap U_i$ is the affine toric variety
\[
\xa\cap U_i =  X_{\cA_i} = \Spec(\C[\sS_i])
\]
where $\cA_i = \cA - m_i$ and $\sS_i = \N\cA_i$.  \qed
\end{proposition}

We also note that the results of Chapter~\ref{affine} imply that the
torus of $X_{\cA_i}$ has character lattice $\Z\cA_i$.  Yet the torus
is $T_{\xa}$, which has character lattice $\Z'\!\cA$ by
Proposition~\ref{2.torusxa}.  These are consistent since $\Z\cA_i =
\Z'\!\cA$ for all $i$.

Besides describing the affine pieces $\xa\cap U_i$ of $\xa \subset
\P^{s-1}$, we can also describe how they patch together.  In other
words, we can give a completely toric description of the inclusions
\[
\xa\cap U_i \ \supseteq \ \xa\cap U_i \cap U_j \ \subseteq \ \xa\cap
U_j
\]
when $i \ne j$.  For instance, $U_i \cap U_j$ consists of those points
of $\xa\cap U_i$ where $x_j/x_i \ne 0$.  In terms of $\xa\cap U_i =
\Spec(\C[\sS_i])$, this means those points where $\chi^{m_j-m_i} \ne
0$.  Thus
\begin{equation}
\label{2.xuiujsg}
\xa\cap U_i \cap U_j = \Spec\big(\C[\sS_i]\big)_{\chi^{m_j-m_i}} =
\Spec\big(\C[\sS_i]_{\chi^{m_j-m_i}}\big).
\end{equation}
so that if we  set $m = m_j-m_i$, so that the inclusion $\xa \cap U_i
\cap U_j \subseteq \xa \cap U_i$ can be written
\begin{equation}
\label{2.xapatching}
\Spec(\C[\sS_i])_{\chi^{m}} \subseteq \Spec(\C[\sS_i]).
\end{equation}
This looks very similar to the inclusion constructed in
\eqref{1.faceopen} of Chapter~\ref{affine} using faces of cones.  We
will see in \S3 that this is no accident.

We next say a few words about how the polytope $P = \Conv(\cA) \subseteq
\mr$ relates to $\xa$.  As we will learn in \S2, the dimension of $P$
is the dimension of the smallest affine subspace of $\mr$ containing
$P$, which is the same as the smallest affine subspace of $\mr$
containing $\cA$.  It follows from Proposition~\ref{2.torusxa} that
\[
\dim \xa = \dim P.
\]
Furthermore, the vertices of $P$ give an especially efficient affine
covering of $\xa$.

\begin{proposition}
\label{2.affinecover}
Given $\cA = \{m_1,\dots,m_s\} \subseteq M$, let $P = \Conv(\cA)
\subseteq \mr$ and set $J = \big\{j \in \{1,\dots,s\} \mid m_j \text{ is
a vertex of } P\big\}$.  Then
\[
\xa = \bigcup_{j \in J} \xa \cap U_j.
\]
\end{proposition}

\begin{proof}
We will prove that if $i \in \{1,\dots,s\}$, then $\xa \cap U_i
\subseteq \xa \cap U_j$ for some $j \in J$.  The discussion of
polytopes from \S2 below implies that
\[
P\cap M_Q = \big\{ {\textstyle\sum_{j \in J}} r_j m_j \mid r_j \in
\Q_{\ge 0},\ {\textstyle \sum_{j\in J}} r_j = 1\big\}.
\]
Given $i \in \{1,\dots,s\}$, we have $m_i \in P\cap M$, so that $m_i$
is a convex $\Q$-linear combination of the vertices $m_j$.  Clearing
denominators, we get integers $k >0$ and $k_j \ge 0$ such that
\[
km_i = {\textstyle \sum_{j\in J}} k_j m_j, \quad 
{\textstyle \sum_{j\in J}} k_j = k.
\]
Thus $\sum_{j\in J} k_j(m_j - m_i) = 0$, which implies $m_i - m_j \in
S_i$ when $k_j > 0$.  Fix such a $j$.  Then $\chi^{m_j - m_i}
\in \C[\sS_i]$ is invertible, so $\C[\sS_i]_{\chi^{m_j - m_i}} =
\C[\sS_i]$.  By~\eqref{2.xuiujsg}, $\xa\cap U_i \cap U_j =
\Spec(\C[\sS_i]) = \xa\cap U_i$, giving $\xa\cap U_i \subseteq
\xa\cap U_j$.
\end{proof}

We conclude with an application to projective normality.  An
irreducible variety $V \subseteq \P^n$ is called \emph{projectively
normal} if its affine cone $\widehat{V} \subseteq \C^{n+1}$ is normal.
A projectively normal variety is always normal
(Exercise~\ref{2.1.projnormalexer}).  Here is an example to show that
the converse can fail.

\begin{example}
\label{2.notprojnormal} 
Let $\cA \subseteq \Z^2$ consist of the columns of the matrix
\[
\begin{pmatrix} 4 & 3 & 1 & 0\\ 0 & 1 & 3 & 4\end{pmatrix},
\]
giving the Laurent monomials $s^4,s^3t,st^3,t^4$.  The polytope $P =
\Conv(\cA)$ is the line segment connecting $(4,0)$ and $(0,4)$, with
vertices corresponding to $s^4$ and $t^4$.  The affine piece of $\xa$
corresponding to $s^4$ has coordinate ring
\[
\C[s^3t/s^4,st^3/s^4,t^4/s^4] = \C[t/s,(t/s)^3,(t/s)^4] = \C[t/s],
\]
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 $\xa$ by
Proposition~\ref{2.affinecover}, so that $\xa$ is normal.

Since $(1,1,1,1)$ is in the row space of the matrix, $\ya$ is the
affine cone of $\xa$ by Proposition~\ref{2.whenhomo}.  The affine
variety $\ya$ is not normal by Example~\ref{1.nonnormalquartic} of
Chapter~\ref{affine}, so that $\xa$ is normal but not projectively
normal.\eoe
\end{example}

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~\ref{normal}, we will see that Example~\ref{2.notprojnormal} is
fully rigorous.  

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

\exercises

\small

\begin{exercise}
\label{2.1.1}
Consider the set $\cP_3 \subseteq \Z^{3\times3}$ of $3\times3$
permutation matrices defined in Example~\ref{2.perm1}.
\begin{thmenum}
\item Prove the claim made in Example~\ref{2.perm1} that three of the
  permutation matrices sum to the other three and use this to explain
  why $x_{123}x_{231}x_{312} - x_{132}x_{321}x_{213} \in
  \I(X_{\cP_3})$.
\item Show that $\dim X_{\cP_3} = 4$ by computing $\Z'\!\cP_3$.
\item Conclude that $\I(X_{\cP_3}) = \langle x_{123}x_{231}x_{312} -
  x_{132}x_{321}x_{213} \rangle$.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{2.1.2}
Let $\cA \subseteq \Z^n$ consist of the columns of an $n\times s$ matrix
with integer entries.  Prove that the conditions of
Proposition~\ref{2.whenhomo} are equivalent to the assertion that
$(1,\dots,1) \in \Z^s$ lies in the row space of the matrix over $\R$
or $\Q$.
\end{exercise}

\begin{exercise}
\label{2.1.dimzp}
Given a finite set $\cA \subseteq M$, prove that the rank of $\Z'\!\cA$
equals the dimension of the smallest affine subspace (over $\Q$ or
$\R$) containing $\cA$. 
\end{exercise}

\begin{exercise}
\label{2.1.zprimeaexer}
Verify the claims made in Example~\ref{2.zprimeaex}.  Also compute
$\I(\ya)$ and check that it is not homogeneous.
\end{exercise}

\begin{exercise}
\label{2.1.projnormalexer}
Let $V \subseteq \P^n$ be projectively normal.  Use
\eqref{2.affinepiece} to prove that the affine pieces $V \cap U_i$ of
$V$ are normal.
\end{exercise}

\begin{exercise}
\label{2.1.invariant}
Fix a finite subset $\cA \subseteq M$.  Given $m \in M$, let $m + \cA =
\{m+m' \mid m' \in \cA\}$.  This is the \emph{translate} of $\cA$ by
$m$. 
\begin{thmenum}
\item Prove that $\cA$ and its translate $m + \cA$ give the same
  projective toric variety, i.e., $\xa = X_{m + \cA}$.
\item Give an example to show that the affine toric varieties $\ya$
  and $Y_{m + \cA}$ can differ.  Hint:\ Pick $\cA$ so that it lies in
  an affine hyperplane not containing the origin.  Then translate
  $\cA$ to the origin.
\end{thmenum}
\end{exercise}

\begin{exercise}
In Proposition~\ref{2.whenhomo}, give a direct proof that (d)
$\Rightarrow$ (c). 
\end{exercise}

\normalsize

\section{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.

\subsection*{\emph{Polytopes}} Recall from Chapter~\ref{affine} that
a polytope $P \subseteq \mr$ is the convex hull of a finite set $S
\subseteq \mr$, 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 \cite{bronsted,grunbaum}.

The \emph{dimension} of a polytope $P \subseteq \mr$ is the dimension
of the smallest affine subspace of $\mr$ containing $P$.  Given a
nonzero vector $u$ in the dual space $\nr$ and $b \in \R$, we get the
\emph{affine hyperplane} $H_{u,b}$ and \emph{closed half-space}
$H^+_{u,b}$ defined by
\[
H_{u,b} = \{m \in \mr \mid \langle m,u\rangle = b\}
\quad\text{and}\quad H^+_{u,b} = \{m \in \mr \mid \langle m,u\rangle \ge
b\}.
\]
A subset $Q \subseteq P$ is a \emph{face} of $P$ if there are $u \in
\nr \setminus \{0\}$ and $b \in \R$ such that
\[
Q = H_{u,b} \cap P \ \text{ and } \ P \subseteq H^+_{u,b}.
\]
We say that $H_{u,a}$ is a \emph{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 $u$.
\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.07in]{ch2fig1.ps}
\end{center}
\caption{A polygon $P$ and four of its supporting lines}  
\end{figure}

We also regard $P$ as a face of itself.  Every face of $P$ is again a
polytope, and if $P = \Conv(S)$ and $Q = H_{u,b} \cap P$ as above,
then $Q = \Conv(S \cap H_{u,b})$.  Faces of $P$ of special interest
are \emph{facets}, \emph{edges} and \emph{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.

\begin{proposition}
\label{2.faceprops}
Let $P \subseteq \mr$ be a polytope.
\begin{thmenum}
\item $P$ is the convex hull of its vertices.  
\item If $P = \Conv(S)$, then every vertex of $P$ lies in $S$.
\item If $Q$ is a face of $P$, then the faces of $Q$ are precisely the
  faces of $P$ lying in $Q$.
\item Every proper face $Q$ is the intersection of the
  facets $F$ containing $Q$.\qed
\end{thmenum}
\end{proposition}

A polytope $P \subseteq \mr$ can also be written as a finite
intersection of closed half-spaces.  The converse is true provided the
intersection is bounded.  In other words, if an intersection
\[
P = \bigcap_{i=1}^s H^+_{u_i,b_i} 
\]
is bounded, then $P$ is a polytope.  Here is a famous example.

\begin{example}
\label{2.birkhoff}
A $d\times d$ matrix $M \in \R^{d\times d}$ is
\emph{doubly-stochastic} if it has nonnegative entries and its row and
sum columns are all $1$.  If we regard $\R^{d\times d}$ as the
affine space $\R^{d^2}$ with coordinates $a_{ij}$, then the set
$\cM_d$ of all doubly-stochastic matrices is defined by the
inequalites
\begin{align*}
{} &a_{ij} \ge 0\quad\qquad\hskip73pt\, (\text{all }i,j)\\ {}
&{\textstyle\sum_{i=1}^d} a_{ij} \ge 1,\,\ \ {\textstyle\sum_{i=1}^d}
a_{ij} \le 1\quad (\text{all }j)\\ {} &{\textstyle\sum_{j=1}^d} a_{ij}
\ge 1,\ \ {\textstyle\sum_{j=1}^d} a_{ij} \le 1\quad (\text{all }i).
\end{align*}
(We use two inequalities to get one equality.)  These inequalities
easily imply that $0 \le a_{ij} \le 1$ for all $i,j$, so that $\cM_d$ is
bounded and hence is a polytope.  

Birkhoff and Von Neumann proved independently that the vertices of
$\cM_d$ are the $d!$ permutation matrices and that $\dim \cM_d =
(d-1)^2$.  In the literature, $\cM_d$ has various names, including the
\emph{Birkhoff polytope} and the \emph{transportation polytope}.  See
\cite[p.\ 20]{ziegler} for more on this interesting polytope.\eoe
\end{example}

When $P$ has maximal dimension, i.e., $\dim P = \dim \mr$, its
presentation as an intersection of closed half-spaces has an
especially nice form because each facet $F$ has a \emph{unique}
supporting affine hyperplane.  We write the supporting affine
hyperplane and corresponding closed half-space as
\[
H_F = \{m \in \mr \mid \langle m,u_F\rangle = -a_F\}
\ \ \text{and}\ \ H_F^+ = \{m \in \mr \mid \langle m,u_F\rangle \ge
-a_F\},
\]
where $(u_F,a_F) \in \nr \times \R$ is unique up to multiplication by
a positive real number.  We call $u_F$ an \emph{inward-pointing facet
normal} of the facet $F$.  It follows that 
\begin{equation}
\label{2.polyhs1}
P = \bigcap_{F \text{ facet}} H_F^+ = \{m \in \mr \mid \langle
m,u_F\rangle \ge -a_F \text{ for all facets } F \text{ of } P\}.
\end{equation}
In Figure~1, the supporting lines plus arrows determine the supporting
half-planes whose intersection is the polygon $P$.  We write
\eqref{2.polyhs1} with minus signs in order to simplify formulas in
later chapters.

Here are some important classes of polytopes.

\begin{definition}
\label{2.typespoly}
Let $P \subseteq \mr$ be a polytope of dimension $d$.
\begin{thmenum}
\item $P$ is a \emph{\bfseries simplex} or \emph{\bfseries
d-simplex} if it has $d+1$ vertices.
\item $P$ is \emph{\bfseries simplicial} if every facet of $P$ is
  simplicial. 
\item $P$ is \emph{\bfseries simple} if every vertex is the
  intersection of precisely $d$ facets.  
\end{thmenum}
\end{definition}

Examples include the Platonic solids in $\R^3$:
\begin{coxitem}
\item A tetrahedron is a $3$-simplex.
\item The octahedron and icosahedron are simplicial since their facets
  are triangles.
\item The cube and dodecahedron are simple since three facets meet at
  every vertex.
\end{coxitem}  

Polytopes $P_1$ and $P_2$ are \emph{combinatorially equivalent} if
there is a bijection
\[
\{\text{faces of } P_1\} \simeq \{\text{faces of } P_2\}
\]
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.

\subsection*{\emph{Sums, Multiples, and Duals}} Given a polytope $P =
\Conv(S)$, its multiple $r P = \Conv(rS)$ is again a polytope for any
$r \ge 0$.  If $P$ is defined by the inequalities
\begin{align*}
\langle m,u_i \rangle &\ge a_i,\quad 1 \le i \le s,\\
\intertext{then $rP$ is given by}
\langle m,u_i \rangle &\ge ra_i,\quad 1 \le i \le s.
\end{align*}
In particular, when $P$ has maximal dimension, then $P$ and $rP$ have
the same inward-pointing facet normals.  

The \emph{Minkowski sum} of subsets $A_1,A_2 \subseteq \mr$ is
\[
A_1+A_1 = \{a_1+a_2 \mid a_1 \in A_1, a_2 \in A_2\}.
\]
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 \emph{dual} or \emph{polar} polytope
\[
P^\circ = \{ u \in \nr \mid \langle m,u\rangle \ge -1 \text{ for all }
u \in P\}.
\]
Figure~2 shows an example of this in the plane.  
\begin{figure}[hb]
\begin{center}
\includegraphics[height=1.5in]{ch2fig2.ps}
\end{center}
\caption{A polygon $P$ and its dual $P^\circ$ in the plane}  
\end{figure}

\bigbreak\goodbreak

When we write $P= \{m \in \mr \mid \langle m,u_F\rangle \ge -a_F, F
\text{ facet}\}$ as in \eqref{2.polyhs1}, we have $a_F > 0$ for all
$F$ since $0$ is in the interior.  Then $P^\circ$ is the convex hull
of the vectors $(1/a_F)u_F \in \nr$ (Exercise~\ref{2.2.dual}).  We
also have $(P^\circ)^\circ = P$ in this situation.

\subsection*{\emph{Lattice Polytopes}} Now let $M$ and $N$ be dual
lattices with associated vector spaces $\mr$ and $\nr$.  A
\emph{\bfseries lattice polytope} $P \subseteq \mr$ is the convex hull
of a finite set $S \subseteq M$.  It follows easily that a polytope in
$\mr$ is a lattice polytope if and only if its vertices lie in $M$
(Exercise~\ref{2.2.lattvert}).  

\begin{example}
\label{2.lattsimpl}
The \emph{standard $n$-simplex} in $\R^n$ is
\[
\Delta_n = \Conv(0,e_1,\dots,e_n).
\]
\begin{figure}[ht]
\begin{center}
\includegraphics[height=2.65in]{ch2fig3.ps}
\end{center}
\caption{The simplex $P = \Conv(0,e_1,e_2,e_1+e_2+3e_3) \subseteq \R^3$}  
\end{figure}
Another simplex in $\R^3$ is $P = \Conv(0,e_1,e_2,e_1+e_2+3e_3)$,
shown in Figure~3.
The lattice polytopes $\Delta_3$ and $P$ are combinatorially
equivalent but will give very different projective toric
varieties.\eoe
\end{example}

\begin{example}
\label{2.birkhoff2}
The Birkhoff polytope defined in Example~\ref{2.birkhoff} is a lattice
polytope relative to the lattice of integer matrices $\Z^{d\times d}$
since its vertices are the permutation matrices, whose entries are all
$0$ or $1$.\eoe
\end{example}

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~\ref{2.2.lattvert}).
Furthermore, every lattice polytope is an intersection of closed
half-spaces defined over $M$, i.e., $P = \bigcap_{i=1}^s
H^+_{u_i,b_i}$ where $u_i \in N$ and $b_i \in \Z$.

When a lattice polytope $P$ has maximal dimension, the facet
presentation given in \eqref{2.polyhs1} 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 $\nr$.  Let $u_F$ denote the unique ray
generator.  The corresponding $a_F$ is integral since $\langle
m,u_F\rangle = -a_F$ when $m$ is a vertex of $F$.  It follows that
\begin{equation}
\label{2.polyhs2}
P = \{m \in \mr \mid \langle m,u_F\rangle \ge -a_F \text{ for
all facets } F \text{ of } P\}.
\end{equation}
is the \emph{unique} facet presentation of the lattice polytope $P$.  

\begin{example}
\label{2.reflexive1}
Conisder the square $P = \Conv(\pm e_1 \pm e_2) \subseteq \R^2$.  The
facet normals of $P$ are $\pm e_1$ and $\pm e_2$ and the facet
presentation of $P$ is given by the inequalities
\begin{align*}
\langle m,\pm e_1 \rangle &\ge -1\\
\langle m,\pm e_2 \rangle &\ge -1.
\end{align*}
Since the $a_F$ are all equal to $1$, it follows that $P^\circ =
\Conv(\pm e_1, \pm e_2)$ is also a lattice polytope.  The polytopes
$P$ and $P^\circ$ are pictured in Figure~2.
\end{example}

It is rare that the dual of a lattice polytope is a lattice
polytope---this is related to the \emph{reflexive polytopes} that will
be studied later in the book.  

\begin{example}
\label{2.badsimplex}
The $3$-simplex $P = \Conv(0,e_1,e_2,e_1+e_2+3e_3) \subseteq \R^3$
pictured in Example~\ref{2.lattsimpl} has facet presentation
\begin{align*}
\langle m,e_3 \rangle &\ge 0\\
\langle m,3e_1-e_3 \rangle &\ge 0\\
\langle m,3e_2-e_3 \rangle &\ge 0\\
\langle m,-3e_1-3e_2+e_3 \rangle &\ge -3
\end{align*}
(Exercise~\ref{2.2.nonnorsimplexer}).  However, if we replace $-3$
with $-1$ in the last inequality, we get integer inequalities that
define $(1/3)P$, which is \emph{not} a lattice polytope.  \eoe
\end{example}

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 $\ge 8$  not
combinatorially equivalent to any lattice polytope.  An example is
described in \cite[Example~6.21]{ziegler}.

\subsection*{\emph{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~\ref{2.badsimplex}
has only four lattice points (its vertices), which implies that the
projective toric variety $X_{P\cap \Z^3}$ is just $\P^3$
(Exercise~\ref{2.2.nonnorsimplexer}).

We will explore two notions of what it means for a lattice polytope to
``have enough lattice points.''  Here is the first.

\begin{definition}
\label{2.normaldef}
A lattice polytope $P \subseteq \mr$ is \emph{\bfseries normal} if
\[
(kP)\cap M + (\ell P)\cap M = ((k+\ell)P)\cap M.
\]
for all $k,\ell \in \N$.
\end{definition}

The inclusion $(kP)\cap M + (\ell P)\cap M \subseteq ((k+\ell)P)\cap
M$ is automatic.  Thus normality means that all lattice points of
$(k+\ell)P$ come from lattice points of $kP$ and $\ell P$.  In
particular, a lattice polytope is normal if and only if
\[
\underbrace{P\cap M + \cdots + P\cap M}_{k \text{ times}} = (kP)\cap
M.
\]
for all integers $k \ge 1$.  In other words, normality says that $P$
has enough lattice points to generate the lattice points in all
integer multiples of $P$.

\begin{definition}
\label{2.smsimpldef}
A simplex $P \subseteq \mr$ is \emph{\bfseries basic} if $P$ has a
vertex $m_0$ such that the differences $m - m_0$, for vertices $m \ne
m_0$ of $P$, form a subset of a $\Z$-basis of $M$.
\end{definition}

This definition is independent of which vertex $m_0 \in P$ is chosen.
The standard simplex $\Delta_n \subseteq \R^n$ is basic, and any basic
simplex is normal (Exercise~\ref{2.2.smsimplexer}).

Here is another interesting class of normal polytopes.

\begin{proposition}
\label{2.polygonnormal}
Every lattice polygon $P \subseteq \R^2$ is normal.
\end{proposition}

\begin{proof}
In Exercise~\ref{2.2.triangulate} you will show that $P$ is a union of
lattice triangles $P = \bigcup_i T_i$ such that each $T_i$ has only
vertices as lattice points.  Such lattice triangles are basic by
Exercise~\ref{2.2.smoothtriangle}.  Take a lattice point $p$ of
$(k+\ell)P$.  This lies in $(k+\ell)T_i$ for some $i$.  The basic
triangle $T_i$ is normal, so that $p$ lies in
\[
(k T_i) \cap \Z^2 + (\ell T_i) \cap \Z^2 \subseteq (k P)\cap\Z^2 +
(\ell P)\cap\Z^2.\qedhere
\]
\end{proof}

The property ``lattice triangles with only vertices as lattice
points are basic and hence normal'' fails in higher dimensions.

\begin{example}
\label{2.nonnorsimplex}
Let $P = \Conv(0,e_1,e_2,e_1+e_2+3e_3) \subseteq \R^3$.  We noted
earlier that the only lattice points of $P$ are its vertices.  But
$P$ is not normal since
\[
e_1+e_2+e_3 = {\textstyle\frac16(0) + \frac13(2e_1) +
\frac13(2e_2) + \frac16(2e_1+2e_2+6e_3)} \in 2P
\]
is not the sum of lattice points of $P$.  It follows that $P$ is not
basic. \eoe
\end{example}

We can also interpret normality in terms of the cone
\[
C_P = \Cone(P\times\{1\}) \subseteq \mr\times\R
\]
introduced in Figure~3 of Chapter~\ref{affine}.  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 $m \in kP$ correspond to points $(m,k) \in C_P\cap
(M\times\Z)$.
\begin{figure}[ht]
\begin{center}
\includegraphics[height=2.75in]{ch2fig4.ps}
\end{center}
\caption{The cone $C_P$ sliced at heights $1$ and $2$}  
\end{figure}
 
In Exercise~\ref{2.2.cphilb} you will show that the semigroup $C_P\cap
(M\times\Z)$ of lattice points in $C_P$ relates to normality as
follows.

\begin{lemma}
\label{2.normalcone}
The lattice polytope $P \subseteq \mr$ is normal if and only if
$(P\cap M)\times\{1\}$ generates the semigroup $C_P\cap
(M\times\Z)$.\qed
\end{lemma}

Here is an important result on normality.

\begin{theorem}
\label{2.kpnormal}
Let $P \subseteq \mr$ be a lattice polytope.  Then there is an integer
$k_0 > 0$ such that $kP$ is normal for all $k \ge k_0$.
\end{theorem}

\begin{proof}
Let $\cH = \{(m_i,h_i)\}$, $i = 1,\dots,s$, be the Hilbert basis of
$C_P\cap (M\times\Z)$.  Note that $m_i \in h_i P$.
In Exercise~\ref{2.2.carefulexer} you will adapt the proof of Gordan's
Lemma (Proposition~\ref{1.gordan} from Chapter~\ref{affine}) to show
that if $\cH$ is the Hilbert basis of the semigroup of lattice points
in a strongly convex cone $\Cone(\cA)$, then the lattice points in the
cone can be written as the union
\begin{equation}
\label{2.careful}
\N\cA \cup {\textstyle\bigcup_{m \in \cH}} \big(m + \N\cA\big).
\end{equation}
Applied to $\cA = (P\cap M)\times\{1\} \subseteq M\times\Z$, this
implies that
\begin{equation}
\label{2.careful2}
C_P\cap(M\times\Z) = \sS \cup {\textstyle\bigcup_{i=1}^s}
\big((m_i,h_i) + \sS\big),
\end{equation}
where $\sS = \N((P\cap M)\times\{1\})$.

Now let $k_0 = \max\{h_i \mid i = 1,\dots,s\}$ and take $k \ge k_0$.
We will prove that $kP$ is normal by showing that
\[
(kP)\cap M + (dkP)\cap M = ((d+1)kP)\cap M
\]
for all $d \in \N$.  It suffices to prove $\supseteq$, so let $m \in
((d+1)kP)\cap M$.  Then $(m,(d+1)k) \in C_P\cap(M\times\Z)$, which
means that $(m,(d+1)k)$ lies in the union \eqref{2.careful2}.  If
$(m,(d+1)k) \in \sS$, then $m$ is a sum of $(d+1)k$ points from $P\cap
M$.  Writing this as a sum of $k$ points plus a sum of $dk$ points
shows that $m \in (kP)\cap M + (dkP)\cap M$.  Also, if
$(m,(d+1)k) \in (m_i,h_i) + \sS$, then
\[
m = m_i + \big(\text{sum of } (d+1)k \! - \! h_i \text{ points of } P\cap
M\big). 
\]
Since $k \ge h_i$, we can write this as 
\[
m = \big(m_i + \text{sum of } k\! - \! h_i \text{ points of } P\cap M\big) +
  \big(\text{sum of } dk \text{ points of } P\cap M\big).
\]
Since $m_i \in h_i P$, it follows that $m \in (kP)\cap M + (dkP)\cap
M$.  
\end{proof}

Here is an example of this result.

\begin{example}
\label{2.nonnorsimplex2}
Consider $P = \Conv(0,e_1,e_2,e_1+e_2+3e_3)$ from
Example~\ref{2.nonnorsimplex}.  The Hilbert basis of
$C_P\cap(M\times\Z)$ 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~\ref{2.2.nonnorsimplexer}).  It follows from the proof of
Theorem~\ref{2.kpnormal} that $2P$ is normal.
\end{example}

In Chapter~\ref{invertible} we will sharpen Theorem~\ref{2.kpnormal}
by showing that if $P$ is an $n$-dimensional lattice polytope, then
$kP$ is normal for all $k \ge n-1$.  This was first proved in
\cite{bgt} and explains why $2P$ is normal in
Example~\ref{2.nonnorsimplex2}.  Also, in Chapter~\ref{coho} we will
show that the number of lattice points in $nP$ is given by a
polynomial in $n$ called the \emph{Ehrhart polynomial}.

\subsection*{\emph{Very Ample Polytopes}} Here is a slightly different
notion of what it means for a polytope to have enough lattice points.

\begin{definition}
\label{2.vadef}
A lattice polytope $P \subseteq \mr$ is \emph{\bfseries very ample} if
for every vertex $m \in P$, the semigroup $\sS_{P,m} = \N(P\cap M -
m)$ generated by $P\cap M - m = \{m' - m \mid m' \in P\cap M\}$ is
saturated in $M$.
\end{definition}

This definition relates to normal polytopes as follows.  

\begin{proposition}
\label{2.norva}
A normal lattice polytope $P$ is very ample.
\end{proposition}

\begin{proof}
Fix a vertex $m_0 \in P$ and take $m \in M$ such that $k m \in
\sS_{P,m_0}$ for some integer $k \ge 1$.  To prove that $m \in
\sS_{P,m_0}$, write $k m \in \sS_{P,m_0}$ as
\[
k m = {\textstyle\sum_{m' \in P\cap M}} a_{m'} (m' - m_0),\quad a_{m'}
\in \N.
\]
Pick $d \in \N$ so that $kd \ge \sum_{m' \in P\cap M} a_{m'}$.  Then
\[
k m + k d m_0 = {\textstyle\sum_{m' \in P\cap M}} a_{m'} m' + \big(kd
- {\textstyle\sum_{m' \in P\cap M}} a_{m'}\big)m_0 \in kd P.
\]
Dividing by $k$ gives $m + d m_0 \in d P$, which by normality implies
that 
\[
m+dm_0 = \sum_{i=1}^d m_i,\quad m_i \in P\cap M \text{ for all } i.
\]
We conclude that $m = \sum_{i=1}^d (m_i-m_0) \in \sS_{P,m_0}$, as desired.
\end{proof}

Combining this with Proposition~\ref{2.polygonnormal} and
Theorem~\ref{2.kpnormal} gives the following.

\begin{corollary}
\label{2.vacor}
Let $P \subseteq \mr$ be a lattice polytope.
\begin{thmenum}
\item If $\dim P = 2$, then $P$ is very ample.
\item There is an integer $k_0 \ge 1$ such that $kP$ is very ample for
  all $k \ge k_0$.\qed
\end{thmenum}
\end{corollary}

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~\ref{2.kpnormal} mentioned
earlier will enable us to take $k_0 = \dim P - 1$ in part~(b) of the
corollary.

The following example taken from \cite[Example 5.1]{bg} shows that
very ample polytopes need not be normal, i.e., the converse of
Proposition~\ref{2.norva} is false.

\begin{example}
\label{2.bgex}
Given $1 \le i < j < k \le 6$, 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
\[
\cA =
\big\{[123],[124],[135],[146],[156],[236],[245],[256],[345],[346]\big\} 
\subseteq \Z^6.
\]
The lattice polytope $P = \Conv(\cA)$ lies in the affine hyperplane of
$\R^6$ where the coordinates sum to $3$.  As explained in \cite{bg},
this configuration can be interpreted in terms of a triangulation of
the real projective plane.  

The points of $\cA$ of $P$ are the only lattice points of $P$
(Exercise~\ref{2.2.bgexer}), so that $\cA$ is the set of vertices of
$P$.  Number the points of $\cA$ as $m_1,\dots,m_{10}$.  Then
\[
(1,1,1,1,1,1) = {\textstyle \frac15} \sum_{i=1}^{10} m_i =
\sum_{i=1}^{10} {\textstyle \frac1{10}}(2m_i)
\]
shows that $v = (1,1,1,1,1,1) \in 2P$.  Since $v$ is not a sum of
lattice points of $P$ (when $[ijk] \in \cA$, the vector with $1$s in
the complementary positions is not in $\cA$), we conclude that $P$ is
not normal.

Showing that $P$ is very ample takes more work.  The first step is to
prove that $\cA\times\{1\} \cup \{(v,2)\} \subseteq \R^6\times\R$ is a
Hilbert basis of the semigroup $C_P \cap \Z^7$, where $C_P \subseteq
\R^6 \times \R = \R^7$ is the cone over $P\times\{1\}$.  We used the
software \textsf{4ti2} \cite{4ti2}.

Now fix $i$ and let $\sS_{P,m_i}$ be the semigroup generated by the
$m_j-m_i$.  Take $m \in \Z^6$ such that $km \in \sS_{P,m_i}$.  As in
the proof of Proposition~\ref{2.norva}, this implies that $m+dm_i \in
dP$ for some $d \in \N$.  Thus $(m+dm_i,d) \in C_P\cap\Z^7$.
Expressing this in terms of the above Hilbert basis easily implies
that
\[
m = a(v - 2m_i) + \sum_{j=1}^{10} a_j(m_j - m_i),\quad a,a_j \in \N.
\]
If we can show that $v - 2m_i \in \sS_{P,m_i}$, then $m \in
\sS_{P,m_i}$ follows immediately and proves that $\sS_{P,m_i}$ is
saturated.  When $i = 1$, one can check that 
\[
v+[123] = [124]+[135]+[236],
\]
which implies that
\[
v-2m_1 = (m_2-m_1)+(m_3-m_1)+(m_6-m_1) \in \sS_{P,m_1}.
\]
One obtains similar formulas for $i = 2,\dots,10$
(Exercise~\ref{2.2.bgexer}), 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
\begin{align*}
& (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).
\end{align*}
Of all $5$-dimensional polytopes whose vertices lie in $\{0,1\}^5$,
this polytope has the most edges, namely 45 (see \cite{aichholzer}).
Since it has $10$ vertices and $45 = \binom{10}{2}$, every pair of
distinct vertices is joined by an edge.  Such polytopes are
\emph{2-neighborly}.\eoe
\end{example}


\exercises

\small

\begin{exercise}
\label{2.2.dual}
Let $P \subseteq \mr$ be a polytope of maximal dimension with the origin
as an interior point.
\begin{thmenum}
\item Write $P = \{m \in \mr \mid \langle m,u_F\rangle \ge -a_F
  \text{ for all facets } F\}$.  Prove that $a_F > 0$ for all $F$ and
  that $P^\circ = \Conv((1/a_F) u_F \mid F \text{ a facet})$. 
\item Prove that the dual of a simplicial polytope is simple and vice
  versa. 
\item Prove that $(rP)^\circ = (1/r)P^\circ$ for all $r > 0$.
\item Use part (c) to construct an example of a lattice polytope whose
  dual is not a lattice polytope.
\end{thmenum}
\end{exercise}
 
\begin{exercise}
\label{2.2.lattvert}
Let $P \subseteq \mr$ be a polytope.
\begin{thmenum}
\item Prove that $P$ is a lattice polytope if and only if the vertices
  of $P$ lie in $M$
\item Prove that $P$ is a lattice polytope if and only if $P$ is the
  convex hull of its lattice points, i.e., $P = \Conv(P\cap M)$.
\item Prove that every face of a lattice polytope is a lattice
  polytope.
\item Prove that Minkowski sums and integer multiples of lattice
  polytopes are again lattice polytopes.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{2.2.nonnorsimplexer}
Let $P = \Conv(0,e_1,e_2,e_1+e_2+3e_3) \subseteq \R^3$ be the simplex
studied in Examples~\ref{2.lattsimpl}, \ref{2.badsimplex},
\ref{2.nonnorsimplex} and \ref{2.nonnorsimplex2}.
\begin{thmenum}
\item Verify the facet presentation of $P$ given in
  Example~\ref{2.badsimplex}. 
\item Show that the only lattice points of $P$ are its vertices.
\item Show that the toric variety $X_{P\cap\Z^3}$ is $\P^3$.
\item Show that the vectors given in Example~\ref{2.nonnorsimplex2}
  form the Hilbert basis of $C_P\cap(M\times\Z)$.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{2.2.smsimplexer}
Recall the definition of basic simplex given in
Definition~\ref{2.smsimpldef}. 
\begin{thmenum}
\item Show that if a simplex satisfies Definition~\ref{2.smsimpldef}
  for one vertex, then it satisfies the definition for all vertices.
\item Show that the standard simplex $\Delta_n$ is basic.
\item Prove that a basic simplex is normal.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{2.2.triangulate}
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 \emph{triangulation} of $P$.
\end{exercise}

\begin{exercise}
\label{2.2.smoothtriangle}
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. 
\end{exercise}

\begin{exercise}
\label{2.2.cphilb}
Prove Lemma~\ref{2.normalcone}.
\end{exercise}

\begin{exercise}
\label{2.2.carefulexer}
Prove \eqref{2.careful}.
\end{exercise}

\begin{exercise}
\label{2.2.bgexer}
Consider the polytope $P = \Conv(\cA)$ from Example~\ref{2.bgex}.
\begin{thmenum}
\item Prove that $\cA$ is the set of lattice points of $P$.
\item Complete the proof begun in the text that $P$ is very ample.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{2.2.simplical}
Prove that every proper face of a simplicial polytope is a simplex.
\end{exercise}

\begin{exercise}
\label{2.2.direct}
In Corollary~\ref{2.vacor} we proved that $kP$ is very ample for $k$
sufficiently large using Theorem~\ref{2.kpnormal} and
Proposition~\ref{2.norva}.  Give a direct proof of this result.
Hint:\ A vertex $m \in P$ gives the cone $C_{P,m}$ generated by the
semigroup $\sS_{P,m}$ defined in Definition~\ref{2.vadef}.  The cone
$C_{P,m}$ is strongly convex since $m$ is a vertex and hence
$C_{P,m}\cap M$ has a Hilbert basis.  Furthermore, $C_{P,m} =
C_{kP,km}$ for all $k \in \N$.  Now argue that when $k$ is large,
$(kP)\cap M - km$ contains the Hilbert basis of $C_{P,m}\cap M$.  A
picture will help.
\end{exercise}

\begin{exercise}
\label{2.2.wtproj}
Fix an integer $a \ge 1$ and consider the $3$-simplex $P =
\Conv(0,ae_1,ae_2,e_3) \subseteq \R^3$.  
\begin{thmenum}
\item Work out the facet presentation of $P$ and verify that the facet
  normals can be labeled so that $u_0 + u_1 + u_2 + au_3 = 0$.
\item Show that $P$ is normal.  Hint:\ Show that $P\cap\Z^3 +
  (kP)\cap\Z^3 = ((k+1)P)\cap\Z^3$.
\end{thmenum}
We will see later that the toric variety of $P$ is the weighted
projective space $\P(1,1,1,a)$.  
\end{exercise}

\normalsize

\section{Polytopes and Projective Toric Varieties}

Our next task is to define the toric variety of a lattice polytope.
As noted in \S2, 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.

\subsection*{\emph{The Very Ample Case}}  Let $P \subseteq \mr$ be a
very ample polytope relative to the lattice $M$, and assume that $\dim
P = \dim \mr = \mathrm{rank}\,M$.  We define
\[
\xp = X_{P\cap M}.
\]
To see what this means concretely, write $P\cap M =
\{m_1,\dots,m_s\}$.  Then $\xp$ is the Zariski closure of the image of
the map $T_N \to \P^{s-1}$ given by
\[
t \longmapsto \big(\chi^{m_1}(t),\dots,\chi^{m_s}(t)\big) \in
\P^{s-1}.
\]
Fix homogeneous coordinates $x_1,\dots,x_s$ for $\P^{s-1}$.

We examine the structure of $X_P \subseteq \P^{s-1}$ using
Propositions~\ref{2.localpieces} and \ref{2.affinecover}.  For each
$m_i \in P\cap M$ consider the semigroup
\[
\sS_i = \N(P\cap M - m_i)
\]
generated by $m_j - m_i$ for $m_j \in P\cap M$.  In $\P^{s-1}$ we have
the affine open subset $U_i \simeq \C^{s-1}$ consisting of those
points where $x_i \ne 0$.  Proposition~\ref{2.localpieces}
showed that the affine open piece $X_P \cap U_i$ of $X_P$ is the
affine toric variety
\[
X_P \cap U_i \simeq \Spec(\C[\sS_i]),
\]
and Proposition~\ref{2.affinecover} showed that 
\[
X_P = \bigcup_{m_i \text{ vertex of } P} X_P \cap U_i.
\]
Here is our first major result about $X_P$.

\begin{theorem}
\label{2.vathm}
Let $X_P$ be the projective toric variety of the very ample polytope
$P \subseteq \mr \simeq \R^n$, and assume that $\dim P = n$.
\begin{thmenum}
\item For each vertex $m_i \in P\cap M$, the affine piece $X_P \cap
  U_i$ is the affine toric variety
\[
X_P \cap U_i = U_{\sigma_i} = \Spec(\C[\sigma_i^\vee\cap M])
\]
where $\sigma_i \subseteq \nr$ is the strongly convex rational
polyhedral cone dual to the cone $\Cone(P\cap M - m_i) \subseteq
\mr$.  Furthermore, $\dim \sigma_i = n$.
\item The torus of $\xp$ has character lattice $M$ and hence is the
  torus $T_N$.
\end{thmenum}  
\end{theorem}

\begin{proof}
Let $C_i = \Cone(P\cap M - m_i)$.  Since $m_i$ is a vertex, it has a
supporting hyperplane $H_{u,a}$ such that $P \subseteq H^+_{u,a}$ and
$P\cap H_{u,a} = \{m_i\}$.  It follows that $H_{u,0}$ is a supporting
hyperplane of $0 \in C_i$ (Exericse~\ref{2.3.vathmexer}), so that
$C_i$ is strongly convex.  It is also easy to see that $\dim C_i =
\dim P$ (Exericse~\ref{2.3.vathmexer}).  It follows that $C_i$ and
$\sigma_i = C_i^\vee$ are strongly convex rational polyhedral cones of
dimension $n$.

We have $\sS_i \subseteq C_i \cap M = \sigma_i^\vee\cap M$.  By
hypothesis, $P$ is very ample, which means that $\sS_i \subseteq M$ is
saturated.  Since $\sS_i$ and $C_i = \sigma_i^\vee$ are both generated
by $P\cap M - m_i$, part (a) of Exercise~\ref{1.3.2} of
Chapter~\ref{affine} implies $\sS_i = \sigma_i^\vee\cap M$.  (This
exercise was part of the proof of the characterization of normal
affine toric vareities given in Theorem~\ref{1.satthm} of
Chapter~\ref{affine}.)  Part (a) of the theorem follows immediately.

For part (b), Theorem~\ref{1.coneafftor} implies that $T_N$ is the
torus of $U_{\sigma_i}$ since $\sigma_i$ is strongly convex.  Then
$T_N \subseteq U_{\sigma_i} = X_P \cap U_i \subseteq \xp$ shows that
$T_N$ is also the torus of $\xp$.
\end{proof}

The affine pieces $X_P \cap U_i$ and $X_P \cap U_j$ intersect in $X_P
\cap U_i \cap U_j$.  In order to describe this intersection carefully,
we need to study how the cones $\sigma_i$ and $\sigma_j$ fit together
in $\nr$.  This leads to our next topic.

\subsection*{\emph{The Normal Fan}} The cones $\sigma_i \subseteq \nr$
appearing in Theorem~\ref{2.vathm} fit together in a remarkably nice
way, giving a structure called the \emph{normal fan of $P$}. 

Let $P \subseteq \mr \simeq \R^n$ 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
\begin{equation}
\label{2.facetpres}
P = \{ m \in \mr \mid \langle m,u_F\rangle \ge -a_F \text{ for all }
F\}.
\end{equation}

A vertex $v \in P$ gives cones
\[
C_v = \Cone(P\cap M - v) \subseteq \mr \quad \text{and} \quad 
\sigma_v = C_v^\vee \subseteq \nr.
\]
(When $v = m_i$, these are the cones $C_i$ and $\sigma_i$ studied
above.)  Faces $Q \subseteq P$ containing $v$ correspond bijectively
to faces $\biq \subseteq C_v$ via the maps
\begin{equation}
\label{2.facespolycone}
\begin{aligned}
Q &\longmapsto \biq = \Cone(Q\cap M - v)\\
\biq &\longmapsto Q = (\biq + v)\cap P
\end{aligned}
\end{equation}
which are inverses of each other.  These maps preserve dimensions,
inclusions, and intersections (Exercise~\ref{2.3.faces}), as
illustrated in Figure~5.
\begin{figure}[ht]
\begin{center}
\includegraphics[height=2.65in]{ch2fig5.ps}
\end{center}
\caption{The cone $C_v$ of a vertex $v \in P$}  
\end{figure}

In particular, all facets of $C_v$ come from facets of $P$ containing
$v$, so that
\[
C_v = \{ m \in \mr \mid \langle m,u_F\rangle \ge 0 \text{ for all } F
\text{ containing } v\}.
\]
By the duality results of Chapter~\ref{affine}, it follows that the
dual cone $\sigma_v$ is given by
\[
\sigma_v = \Cone(u_F \mid F \text{ contains } v).
\]
This construction generalizes to arbitrary faces $Q \subseteq P$ by
setting
\[
\sigma_Q = \Cone(u_F \mid F \text{ contains } Q).
\]
In particular, $\sigma_F$ is the ray generated by $u_F$, and $\sigma_P
= \{0\}$ since the $\{0\}$ is the cone generated by the empty set.

The following easy lemma will be useful.

\begin{lemma}
\label{2.qsq}
Let $Q$ be a face of $P$ and let $H_{u,b}$ be a supporting affine
hyperplane of $P$.  Then $u \in \sigma_Q$ if and only if $Q \subseteq
H_{u,b}\cap P$.
\end{lemma}

\begin{proof}
First suppose that $u \in \sigma_Q$ and write $u= \sum_{Q \subseteq F}
\lambda_F u_F$, $\lambda_F \ge 0$.  Then setting $b = -\sum_{Q
\subseteq F} \lambda_F a_F$ easily implies that $P \subseteq
H^+_{u,b}$ and $Q \subseteq H_{u,b}\cap P$.  Recall that the integers
$a_F$ come from the facet presentation \eqref{2.facetpres}.

Going the other way, suppose that $Q \subseteq H_{u,b}\cap P$.  Take a
vertex $v \in Q$.  Then $P \subseteq H^+_{u,b}$ and $v \in H_{u,b}$
imply that $C_v \subseteq H^+_{u,0}$.  Hence $u \in C_v^\vee =
\sigma_v$, so that
\[
u = {\textstyle \sum_{v \in F}} \lambda_F u_F,\quad \lambda_F \ge 0.
\]
Let $F_0$ be a facet of $P$ containing $v$ but not $Q$, and pick
$p \in Q$ with $p \notin F_0$.  Then $p,v \in Q \subseteq
H_{u,b}$ imply that
\begin{align*}
b &= \langle p,u\rangle = {\textstyle \sum_{v \in F}} \lambda_F\langle
p,u_F\rangle\\
b &= \langle v,u\rangle = {\textstyle \sum_{v \in F}} \lambda_F\langle
v,u_F\rangle = -{\textstyle \sum_{v \in F}}\lambda_F a_F.
\end{align*}
where the last equality uses $\langle v,u_F\rangle = -a_F$ for $v \in
F$.  These equations imply
\[
{\textstyle \sum_{v \in F}} \lambda_F\langle
p,u_F\rangle = -{\textstyle \sum_{v \in F}}\lambda_F a_F.
\]
However, $p \notin F_0$ gives $\langle p,u_{F_0}\rangle > -a_{F_0}$,
and since $\langle p,u_{F}\rangle \ge - a_F$ for all $F$, it follows
that $\lambda_{F_0} = 0$ whenever $Q \not\subseteq F_0$.  This gives
$u \in \sigma_Q$ and completes the proof of the lemma.
\end{proof}

\begin{corollary}
\label{2.qsqcor}
If $Q$ is a face of $F$, then $u_F \in \sigma_Q$ if and only if $Q
\subseteq F$.
\end{corollary}

\begin{proof}
One direction is obvious by the definition of $\sigma_Q$, and the
other direction follows from Lemma~\ref{2.qsq} since $H_{u_F,-a_F}$ is
a supporting affine hyperplane of $P$ with $H_{u_F,-a_F}\cap P = F$.
\end{proof}

We are now ready to explore the combinatorial and geometric properties
of the cones $\sigma_Q$.  We begin with some combinatorial properties.

\bigbreak\goodbreak

\begin{proposition}
\label{2.norfancomb}
Let $Q$ and $Q'$ be faces of a lattice polytope $P \subseteq \mr
\simeq \R^n$ of dimension $n$.  Then:
\begin{thmenum}
\item $Q \subseteq Q'$ if and only if $\sigma_{Q'}\subseteq \sigma_{Q}$.  
\item If $Q \subseteq Q'$, then $\sigma_{Q'}$ is a face of $\sigma_Q$,
  and all faces of $\sigma_Q$ are of this form.
\item $\sigma_Q \cap \sigma_{Q'} = \sigma_{Q''}$, where $Q''$ is the
  smallest face of $P$ containing $Q$ and $Q'$.
\end{thmenum}
\end{proposition}

\begin{proof}
To prove part (a), note that if $Q \subseteq Q'$, then any facet
containing $Q'$ also contains $Q$, which implies $\sigma_{Q'} \subseteq
\sigma_{Q}$.  The other direction follows easily from
Corollary~\ref{2.qsqcor} since every face is the intersection of the
facets containing it by Proposition~\ref{2.faceprops}.

For part (b), fix a vertex $v \in Q$ and note that by
\eqref{2.facespolycone}, $Q$ determines a face $\biq$ of $C_v$.  Using
the duality of Proposition~\ref{1.dualfaceprops} of
Chapter~\ref{affine}, $\biq$ gives the dual face
\[
\biq^* = C_v^\vee\cap \biq^\perp = \sigma_v \cap \biq^\perp
\]
of the cone $\sigma_v$.  Then using $\sigma_v = \Cone(u_F \mid v \in
F)$ and $\biq \subseteq C_v = \sigma_v^\vee$, one obtains
\[
\biq^* = \Cone(u_F \mid v \in F,\ \biq \subseteq H_{u_F,0}).
\]
Since $v \in Q$, the inclusion $\biq \subseteq H_{u_F,0}$ is
equivalent to $Q \subseteq H_{u_F,-a_F}$, which in turn is equivalent
to $Q \subseteq F$.  It follows that
\begin{equation}
\label{2.biqssq}
\biq^* = \Cone(u_F \mid Q \subseteq F) = \sigma_Q,
\end{equation}
so that $\sigma_Q$ is a face of $\sigma_v$, and all faces of
$\sigma_v$ arise in this way.  

In particular, $Q \subseteq Q'$ means that $\sigma_{Q'}$ is also a
face of $\sigma_v$, and since $\sigma_{Q'} \subseteq \sigma_{Q}$ by
part (a), we see that $\sigma_{Q'}$ a face of $\sigma_Q$.
Furthermore, every face of $\sigma_Q$ is a face of $\sigma_v$ by
Proposition~\ref{1.faceprops} of  Chapter~\ref{affine} and hence is of
the form $\sigma_{Q'}$ for some face $Q'$.  Using part (a) again, we
see that  $Q \subseteq Q'$, 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) $\sigma_{Q''}$ is a facet of both $\sigma_Q$ and
$\sigma_{Q'}$.  Thus $\sigma_{Q''} \subseteq \sigma_Q
\cap\sigma_{Q'}$.  

It remains to prove the opposite inclusion.  If $\sigma_Q
\cap\sigma_{Q'} = \{0\} = \sigma_P$, then $Q'' = P$ and we are done.
If $\sigma_Q \cap\sigma_{Q'} \ne \{0\}$, any nonzero $u$ in the
intersection lies in both $\sigma_Q$ and $\sigma_{Q'}$.  The proof of
Proposition~\ref{2.norfangeom} given below will show that $H_{u,b}$ is
a supporting affine hyperplane of $P$ for some $b \in \R$.  By
Lemma~\ref{2.qsq}, $u \in \sigma_Q$ and $u \in \sigma_{Q'}$ imply that
$Q$ and $Q'$ lie in $H_{u,b}\cap P$.  The latter is a face of $P$
containing $Q$ and $Q'$, so that $Q'' \subseteq H_{u,b}\cap P$ since
$Q''$ is the smallest such face.  Applying Lemma~\ref{2.qsq} again, we
see that $u \in \sigma_{Q''}$.
\end{proof}

Proposition~\ref{2.norfancomb} shows that there is a bijective
correspondence between faces and cones.  The proposition also implies
that the cones fit together as follows.

\begin{corollary}
\label{2.norfancombcor}
Let $P \subseteq \mr \simeq \R^n$ be a lattice polytope of dimension
$n$ and set $\Sigma_P = \{\sigma_Q \mid Q \text{ is a face of } Q\}$.
Then:
\begin{thmenum}
\item For all $\sigma_Q \in \Sigma_P$, each face of $\sigma_Q$ is also
  in $\Sigma_P$.
\item The intersection $\sigma_Q \cap \sigma_{Q'}$ of any two cones in
  $\Sigma_P$ is a face of each.
\end{thmenum}
\end{corollary}

In Chapter~\ref{normal}, a collection of strongly convex rational
polyhedral cones satisfying conditions (a) and (b) of
Corollary~\ref{2.norfancombcor} will be called a \emph{fan}.  Since
the cones in the above fan $\Sigma_P$ are built from the
inward-pointing normal vectors $u_F$, we call $\Sigma_P$ the
\emph{normal fan} or \emph{inner normal fan} of $P$.

Before we draw pictures of normal fans, we mention some of the
geometric properties of the cones $\sigma_Q$.

\begin{proposition}
\label{2.norfangeom}
Let $P \subseteq \mr \simeq \R^n$ be a lattice polytope of dimension
$n$ and consider the cones $\sigma_Q$ in the normal fan $\Sigma_P$ of
$P$.  Then:
\begin{thmenum}
\item $\dim Q + \dim \sigma_Q = n$ for all faces $Q$ of $P$.
\item $\nr = \bigcup_{v} \sigma_v = \bigcup_{Q \text{\rm\ face of P}}
\sigma_Q$.
\end{thmenum}
\end{proposition}

\begin{proof} Let $Q$ be a face of $P$ and take a vertex $v$ of $Q$.
By \eqref{2.facespolycone} this gives a face $\biq$ of the cone $C_v$,
which has a dual face $\biq^*$ of the dual cone $C_v^\vee = \sigma_v$.
Since $\biq^* = \sigma_Q$ by \eqref{2.biqssq}, we have
\[
\dim Q + \dim \sigma_Q = \dim \biq + \dim \biq^* = n,
\]
where the first equality uses Exercise~\ref{2.3.faces} and the second
follows from Proposition~\ref{1.dualfaceprops} of
Chapter~\ref{affine}.  This proves part (a).  For part (b), let $u \in
\nr$ be nonzero and set $b = \min\{\langle v,u\rangle \mid v 
\text{ vertex of } P\}$.  Then $P \subseteq H^+_{u,b}$ and $v \in 
H_{u,b}$ for at least one vertex of $P$, so that $u \in \sigma_v$ by
Lemma~\ref{2.qsq}.  To prove the final equality of part (b), note that
every face $Q$ contains a vertex $v$, so that $\sigma_Q$ is a face of
$\sigma_v$ by Proposition~\ref{2.norfancomb}.
\end{proof}

In the terminology of Chapter~\ref{normal}, a fan satisfying the
condition of part (b) of Proposition~\ref{2.norfangeom} is called
\emph{complete}.  Thus the normal fan of a lattice polytope is always
complete.  

\begin{example}
\label{2.hexagon}
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
$v_1,\dots, v_6$, with corresponding cone $\sigma_1,\dots,\sigma_6$ 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 $\sigma_i$ they generate.  On the
right, these cones are assembled at the origin to give the normal fan.
\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.65in]{ch2fig6.ps}
\end{center}
\caption{A lattice hexagon $P$ and its normal fan $\Sigma_P$}  
\end{figure}

Notice how one can read off the structure of $P$ from the normal fan.
For example, two cones $\sigma_i$ and $\sigma_j$ share a ray in
$\Sigma_P$ if and only if the vertices $v_i$ and $v_j$ lie on an edge
of $P$. \eoe
\end{example}

Here is a similar example that illustrates an important property of
normal fans.

\begin{example}
\label{2.p2ex}
The $2$-simplex $\Delta_2 \subseteq \R^2$ has vertices $0,e_1,e_2$.
Let $P = k\Delta_2$ for some positive integer $k$.  Figure~7 shows $P$
and its normal fan $\Sigma_P$.
\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.65in]{ch2fig7.ps}
\end{center}
\caption{The triangle $P = k\Delta_2 \subseteq \R^2$ and its normal
  fan $\Sigma_P$}
\end{figure}
At each vertex $v_i$ of $P$, we have drawn
the normal vectors of the facets containing $v_i$ and shaded the cone
$\sigma_i$ they generate.  The reassembled cones appear on the left
as $\Sigma_P$. \eoe
\end{example}

An important observation is that the normal fan in
Example~\ref{2.p2ex} 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~\ref{2.3.samenf}).

\begin{proposition}
\label{2.norfanunchanged}
Let $P \subseteq \mr \simeq \R^n$ be an $n$-dimensional lattice
polytope.  Then for any lattice point $m \in M$ and any integer $k
\ge1$, the polytopes $m + P$ and $kP$ have the same normal fan as
$P$. \qed
\end{proposition}

Here is an example of the normal fan of a polytope in $\R^3$.

\begin{example}
\label{2.cubedual}
Consider the cube $P \subseteq \R^3$ with vertices
$(\pm1,\pm1,\pm1)$.  The facet normals are $\pm e_1,\pm e_2, \pm e_3$,
and the facet presentation of $P$ is 
\[
\langle m, \pm e_i \rangle \ge -1.
\]
The origin is an interior point of $P$.  By Exercise~\ref{2.2.dual},
the facet normals are the vertices of the dual polytope $P^\circ$, the
octahedron pictured in Figure~8.
\begin{figure}[hb]
\begin{center}
\includegraphics[height=2.4in]{ch2fig8.ps}
\end{center}
\caption{A cube $P \subseteq \R^3$ and its dual octahedron $P^\circ$}  
\end{figure}

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 $P^\circ$. \eoe 
\end{example}

As noted earlier, it is rare that both $P$ and $P^\circ$ are lattice
polytopes.  However, whenever $P \subseteq \mr$ is a lattice polytope
containing $0$ as an interior point, it is still true that maximal
cones of the normal fan $\Sigma_P$ are the cones over the facets of
$P^\circ \subseteq \nr$ (Exercise~\ref{2.3.dualnf}).  

The special behavior of the polytopes $P$ and $P^\circ$ discussed in
Examples~\ref{2.reflexive1} and~\ref{2.cubedual} leads to the
following definition.

\begin{definition}
\label{2.reflexivedef}
An $n$-dimensional lattice polytope $P \subseteq \mr \simeq \R^n$ is
\emph{\bfseries reflexive} if its facet presentation is
\[
P = \{ m \in \mr \mid \langle m,u_F\rangle \ge -1 \text{ for all
  facets } F\}.
\] 
\end{definition}

If $P$ is reflexive, then $0$ is a lattice point of $P$ and and is in
fact the \emph{only} interior lattice point of $P$
(Exercise~\ref{2.3.reflexexiveexer}).  Since $a_F = 1$ for all $F$,
Exercise~\ref{2.2.dual} implies that
\[
P^\circ = \Conv(u_F \mid F \text{ facet of } P).
\]
Thus $P^\circ$ is a lattice polytope and is in fact reflexive
(Exercise~\ref{2.3.reflexexiveexer}).

\bigbreak\goodbreak

We will see later that reflexive polytopes lead to some very
interesting toric varieties that are important for mirror symmetry.
 
\subsection*{\emph{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:
\begin{coxitem}
\item What is the interesection two of the affine pieces
  described in Theorem~\ref{2.vathm}?
\item How do we define the projective toric variety $X_P$ when
  $P$ is not very ample?
\end{coxitem}
As we will see, these questions are linked in a very satisfying manner.  

To study the first question, let $P \subseteq \mr \simeq \R^n$ be an
$n$-dimensional very ample polytope and set $s = |P\cap M|$.  Then
\[
X_P = X_{P\cap M} \subseteq \P^{s-1}.
\]
Label the homogeneous coordinates of $\P^{s-1}$ as $x_m$ for $m \in
P\cap M$.  If $v$ is a vertex of $P$, then $U_v \subseteq \P^{s-1}$ is
the affine open subset where $x_v \ne 0$, and Theorem~\ref{2.vathm}
tells us that
\[
\xp \cap U_v = U_{\sigma_v} = \Spec(\C[\sigma_v^\vee\cap M]).
\]
Thus $\xp \cap U_v$ is the affine toric variety of the cone $\sigma_v$
in the normal fan $\Sigma_P$ of $P$.

\begin{proposition}
\label{2.uvuw}
Let $P \subseteq \mr \simeq \R^n$ be very ample of dimension $n$.  If
$v \ne w$ are vertices of $\xp$ and $Q$ is the smallest face of $P$
containing $v$ and $w$, then
\[
\xp \cap U_v \cap U_w = U_{\sigma_Q} = \Spec(\C[\sigma_Q^\vee\cap M])
\]
and the inclusions 
\[
\xp \cap U_v \supseteq \xp \cap U_v \cap U_w \subseteq \xp \cap U_w
\] 
can be written
\begin{equation}
\label{2.gluing}
U_{\sigma_v} \supseteq (U_{\sigma_v})_{\chi^{w-v}} = U_{\sigma_Q}
= (U_{\sigma_w})_{\chi^{v-w}} \subseteq U_{\sigma_w}.
\end{equation}
\end{proposition}

\begin{proof}
We analyzed the intersection of affine pieces of $X_{P\cap M}$ in
\S1.  Translated to the notation being used here, \eqref{2.xuiujsg}
and \eqref{2.xapatching} imply that
\[
\xp \cap U_v \cap U_w = (U_{\sigma_v})_{\chi^{w-v}} =
(U_{\sigma_w})_{\chi^{v-w}}.
\]
Thus all we need to show is that
\[
(U_{\sigma_v})_{\chi^{w-v}} = U_{\sigma_Q}.
\]
However, we have $w-v \in C_v = \sigma_v^\vee$, so that $\tau =
H_{w-v}\cap \sigma_v$ is a face of $\sigma_v$.  In this situation,
Proposition~\ref{1.facesg} and equation \eqref{1.faceopen} of
Chapter~\ref{affine} imply that
\[
(U_{\sigma_v})_{\chi^{w-v}} = U_\tau.
\]
Thus the proposition will follow once we prove $\tau = \sigma_Q$,
i.e., $H_{w-v}\cap \sigma_v = \sigma_Q$.  Since $\sigma_Q = \sigma_v
\cap \sigma_w$ by Proposition~\ref{2.norfancomb}, it suffices to prove
that
\[
H_{w-v}\cap \sigma_v = \sigma_v \cap \sigma_w.
\]
Let $u \in H_{w-v}\cap \sigma_v$.  If $u \ne 0$, there is $b \in \R$
such $H_{u,b}$ is a supporting affine hyperplane of $P$.  Then $u \in
\sigma_v$ implies $v \in H_{u,b}$ by Lemma~\ref{2.qsq}, so that $w \in
H_{u,b}$ since $u \in H_{w-v}$.  Applying Lemma~\ref{2.qsq} again, we
get $u \in \sigma_w$.  Going the other way, let $u \in \sigma_v \cap
\sigma_w$.  If $u \ne 0$, pick $b \in \R$ as above.  Then $u \in
\sigma_v \cap \sigma_w$ and Lemma~\ref{2.qsq} imply that $v,w \in
H_{u,b}$, from which $u \in H_{w-v}$ follows easily.  This completes
the proof.
\end{proof}

This proposition and Theorem~\ref{2.vathm} have the remarkable result
that the normal fan $\Sigma_P$ completely determines the internal
structure of $\xp$:\ we build $\xp$ from local pieces given by the
affine toric varieties $U_{\sigma_v}$, glued together via
\eqref{2.gluing}.  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.

\begin{definition}
\label{2.xpdef}
Let $P \subseteq \mr \simeq \R^n$ be an $n$-dimensional lattice
polytope.  Then we define 
\[
\xp = X_{(kP)\cap M}
\]
where $k$ is any positive integer such that $kP$ is very ample.
\end{definition}

Such integers $k$ exist by Corollary~\ref{2.vacor}, and if $k$ and
$\ell$ are two such integers, then $kP$ and $\ell P$ have the same
normal fan by Proposition~\ref{2.norfanunchanged}, namely $\Sigma_{kP}
= \Sigma_{\ell P} = \Sigma_P$.  It follows that while $X_{(kP)\cap M}$
and $X_{(\ell P)\cap M}$ lie in different projective spaces, they are
built from the affine toric varieties $U_{\sigma_v}$ glued together
via \eqref{2.gluing}.  Once we develop the language of abstract
varieties in Chapter~\ref{normal}, we will see that $\xp$ is
well-defined as an abstract variety.

We will often speak of $\xp$ without regard to the projective
embedding.  When we want to use a specific embedding, we will say
``$\xp$ is embedded using $kP$'', where we assume that $kP$ is very
ample.  In Chapters~\ref{divisors} and \ref{invertible} we will use
the language of divisors and line bundles to restate this in terms of
a divisor $D_P$ on $\xp$ such that $kD_P$ is very ample precisely when
$kP$ is.

Here is a simple example to illustrate the difference between $\xp$ as
an abstract variety and $\xp$ as sitting is a specific projective
space.

\begin{example}
\label{2.veronese}
Consider the $n$-simplex $\Delta_n \subseteq \R^n$.  We can define
$X_{\Delta_n}$ using any integer $k \ge 1$ since $\Delta_n$ is normal
and hence very ample.  The lattice points in $k\Delta_n$ correspond to
the $s_k = \binom{n+k}{k}$ monomials of $\C[t_1,\dots,t_n]$ of total
degree $\le n$.  This gives an embedding $X_{\Delta_n} \subseteq
\P^{s_k-1}$.  When $k = 1$, $\Delta_n\cap\Z^n = \{0,e_1,\dots,e_n\}$,
which easily implies that
\[
X_{\Delta_n} = \P^n.
\]
The normal fan of $\Delta_n$ is described in
Exercise~\ref{2.3.veroneseexer}.  For an arbitrary $k \ge 1$, we can
regard $X_{\Delta_n} \subseteq \P^{s_k-1}$ as the image of the map
\[
\nu_k : \P^n \longrightarrow \P^{s_k-1}
\]
defined using all monomials of total degree $k$ in $\C[x_0,\dots,x_n]$
(Exercise~\ref{2.3.veroneseexer}).  It follows that this map is an
embedding, usually called the \emph{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 $f \in \C[x_0,\dots,x_n]$ be nonzero and
homogeneous of degree $k$ and write $f = \sum_{|\a| = k} c_\a x^\a$.
We write the homogeneous coordinates of $\P^{s-1}$ as $y_\a$ for $|\a|
= k$.  Then $L = \sum_{|\a| = k} c_\a y_\a$ is a nonzero linear form
in the variables $y_\a$, so that $\P^{s_k-1} \setminus \V(L)$ is a copy
of $\C^{s-1}$ (Exercise~\ref{2.3.veroneseexer}).  If follows that
\[
\P^n \setminus \V(f) \simeq \nu_k(\P^n) \cap \big(\P^{s_k-1} \setminus
\V(L)\big)
\]
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 $U_i =
\P^n \setminus \V(x_i)$ considered earlier in the chapter.\eoe 
\end{example}

When we explain the Proj construction $\P^n$ later in the book, we
will see the intrinsic reason why $\P^n \setminus \V(f)$ is an affine
open subset of $\P^n$.

\begin{example}
\label{2.rnscroll}
The $2$-dimensional analog of the rational normal curve $C_n$ is the
\emph{rational normal scroll} $S_{a,b}$, which is the toric variety of
the polygon
\[
P_{a,b} = \Conv(0, ae_1, e_2, be_2 + e_2) \subseteq \R^2,
\]
where $a,b \in \N$ satisfy $1 \le a \le b$.  The polygon $P = P_{2,4}$
and its normal fan are pictured in Figure~9.
\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.6in]{ch2fig9.ps}
\end{center}
\caption{The polygon of a rational normal scroll and its normal fan}  
\end{figure}

In general, the polygon $P_{a,b}$ has $a+b+2$ lattice points and gives
the map
\[
(\Cstar)^2 \longrightarrow \P^{a+b+1}, \quad (s,t) \mapsto
(1,s,s^2,\dots,s^a,t,st,s^2t,\dots,s^bt)
\]
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
\[
\C \times \P^1 \longrightarrow \P^{a+b+1},\quad (s,\lambda,\mu)
\mapsto (\lambda,s\lambda, s^2\lambda,\dots,
s^a\lambda,\mu,s\mu,s^2\mu,\dots,s^b\mu).
\]
When $(\lambda,\mu) = (1,0)$, the map is $s \mapsto
(1,s,s^2,\dots,s^a,0,\dots,0)$, which is the rational normal curve
$C_a$ mapped to the first $a+1$ coordinates of $\P^{a+b+1}$.  In the
same way, $(\lambda,\mu) = (0,1)$ 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 $(\lambda,\mu) \in \P^1$ 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 \emph{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 $r \in \N$, it
follows that as abstract toric varieties, we have
\[
X_{P_{1, r+1}} = X_{P_{2,r+2}} = X_{P_{2,r+2}} = \cdots
\]
since they are all constructed from the same normal fan.  In
Chapter~\ref{normal}, we will see that this is the Hirzebruch surface
$\cH_r$.

But if we think of the projective surface $S_{a,b} \subseteq
\P^{a+b+1}$, 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
$x_0,\dots,x_a,y_0,\dots,y_b$ and consider the $2\times (a+b)$ matrix
\[
\left(\begin{matrix} x_0 & x_1 & \cdots & x_{a-1} \\
x_1 & x_1 & \cdots & x_{a}\end{matrix}\right|\left.
\begin{matrix} y_0 & y_1 & \cdots & y_{b-1} \\
y_1 & y_1 & \cdots & y_{b}\end{matrix}\right).
\]
One can show that $I(S_{a,b} \subseteq \C[x_0,\dots,x_a,y_0,\dots,y_b]$
is generated by the $2\times2$ minors of this matrix (see
\cite[Exercise~9.11]{harris}, for example).  \eoe
\end{example}

Example~\ref{2.rnscroll} is another example of a determinantal variety,
as is the rational normal curve from Example~\ref{2.rnccurve}.  Note
that the rational normal curve $C_n$ comes from the polytope $[0,n] =
n\Delta_1$, where the underlying toric variety is just $\P^1$.

\exercises

\small

\begin{exercise}
\label{2.3.vathmexer}
This exercise will use the same notation as the proof of
Theorem~\ref{2.vathm}. 
\begin{thmenum}
\item Let $H_{u,a}$ be a supporting hyperplane of a vertex $m_i \in
P$.  Prove that $H_{u,0}$ is a supporting hyperplane of $0 \in C_i$
\item Prove that $\dim C_i = \dim P$.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{2.3.faces}
Consider the maps defined in \eqref{2.facespolycone}.
\begin{thmenum}
\item 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 $v$.
\item Prove that these maps preserve dimensions, inclusions, and
intersections. 
\item Explain how this exercise relates to Exercise\ref{2.3.vathmexer}.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{2.3.samenf}
Prove Proposition~\ref{2.norfanunchanged}.
\end{exercise}

\begin{exercise}
\label{2.3.dualnf}
Let $P \subseteq \mr \simeq \R^n$ be an $n$-dimensional lattice
polytope containing $0$ as an interior point, and let $P^\circ \subset
\nr$ be its dual polytope.  Prove that the normal fan $\Sigma_P$
consists of the cones over the faces of $P^\circ$.  Hint:\
Exercise~\ref{2.2.dual} will be useful.
\end{exercise}

\begin{exercise}
\label{2.3.reflexexiveexer}
Let $P \subseteq \mr$ be a reflexive polytope.
\begin{thmenum}
\item Prove that $0$ is the only interior lattice point of $P$.
\item Prove that $P^\circ \subset \nr$ is reflexive.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{2.3.veroneseexer}
This exercise is concerned with Example~\ref{2.veronese}
\begin{thmenum}
\item Let $e_1,\dots,e_n$ be the standard basis of $\R^n$.  Prove that the
normal fan of the standard $n$-simplex consists of the cones
$\Cone{S}$ for all proper subsets $S \subseteq \{e_0,e_1,\dots,e_n\}$.
Draw pictures of the normal fan for $n = 1, 2, 3$.
\item For an integer $k \ge 1$, show that the variety  $X_{k\Delta_n}
\subseteq \P^{s_k-1}$ is given by the map $\nu_k : \P^n
\longrightarrow \P^{s_k-1} $defined using all monomials of total
degree $k$ in $\C[x_0,\dots,x_n]$.
\end{thmenum}
\end{exercise}

% exercises not mentioned in text go here

\begin{exercise}
\label{2.3.gencqsq}
Let $P \subseteq \mr \simeq \R^n$ be an $n$-dimensional lattice
polytope and let $Q \subseteq P$ be a face.  Prove the
following intrinsic description of cone $\sigma_Q \in \Sigma_P$:
\[
\sigma_Q - \{u \in \nr \mid \langle m,u\rangle \le \langle m',u\rangle
\text{ for all } m \in Q,\ m' in P\}.
\] 
\end{exercise}

\begin{exercise}
\label{2.3.rectangle}
Prove that all lattice rectangles in the plane with edges parallel to
the coordinate axes have the same normal fan.
\end{exercise}


\normalsize


\section{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.  

\subsection*{\emph{Normality}}  For the toric variety $\xp$ of a
polytope, we have the following basic result on normality.

\begin{theorem}
\label{2.normalthm}
Let $P \subseteq \mr \simeq \R^n$ be an $n$-dimensional lattice
polytope.  Then:
\begin{thmenum}
\item $X_P$ is normal.
\item $X_P$ is projectively normal under the embedding given
by $kP$ if and only if $kP$ is normal.
\end{thmenum}
\end{theorem}

\begin{proof}
Part (a) is immediate since $X_P$ is the union of affine pieces
$U_{\sigma_v}$, $v$ a vertex of $P$, and $U_{\sigma_v}$ is normal by
Theorem~\ref{1.satthm} of Chapter~\ref{affine}.  In
Chapter~\ref{normal} we will give an intrinsic definition of normality
that will make this argument completely rigorous.

For part (b), the discussion following \eqref{2.ax1} shows that the
projective embedding of $\xp$ given by $X_{(kP)\cap M}$ has affine
cone given by $Y_{((kP)\cap M)\times\{1\}}$.  By
Theorem~\ref{1.satthm} of Chapter~\ref{affine}, this is normal if and
only if the semigroup $\N((kP)\cap M)\times\{1\}$ is saturated in
$M\times\Z$, and since $((kP)\cap M)\times\{1\}$ generates the cone
$C_P$, this is equivalent to saying that $(kP)\cap M)\times\{1\}$
generates the semigroup $C_P\cap(M\times\Z)$.  Then we are done by
Lemma~\ref{2.normalcone}.
\end{proof}

\subsection*{\emph{Smoothness}} Given the results of
Chapter~\ref{affine}, the smoothness of $\xp$ is equally easy to
determine.  We need one definition.

\bigbreak\goodbreak

\begin{definition}
\label{2.smpolydef}
Let $P \subseteq \mr$ be a lattice polytope.
\begin{thmenum}
\item 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 $\Cone(E - v)$.
\item $P$ is \emph{\bfseries 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 $\dim P = \dim \mr$, then
the vectors $w_E - v$ form a basis of $M$.
\end{thmenum}
\end{definition}

We can now characterize when $\xp$ is smooth.

\begin{theorem}
\label{2.smpolythm}
Let $P \subseteq \mr \simeq \R^n$ be an $n$-dimensional lattice
polytope.  Then the following are equivalent:
\begin{thmenum}
\item $X_P$ is a smooth projective variety.
\item $\Sigma_P$ is a smooth fan, meaning that every cone in
  $\Sigma_P$ is smooth in the sense of Definition~\ref{1.regsimp}.
\item $P$ is a smooth polytope.
\end{thmenum}
\end{theorem}

\begin{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 $U_{\sigma_v}$ is smooth for every vertex $v$ of $P$, and
$U_{\sigma_v}$ is smooth if and only if $\sigma_v$ is smooth.  Since
faces of smooth cones are smooth and $\Sigma_P$ consists of the
$\sigma_v$ and their faces, the equivalence (a) $\Leftrightarrow$ (b)
follows immediately.

For (b) $\Leftrightarrow$ (c), first observe that $\sigma_v$ is smooth
if and only if it dual $C_v = \sigma_v^\vee$ is smooth.  The
discussion following \eqref{2.facespolycone} makes it easy to see that
the ray generators of $C_v$ are the vectors $w_E - v$ from
Definition~\ref{2.smpolydef}.  It follows immediately that $P$ is
smooth if and only if $C_v$ is smooth for every vertex $v$, and we are
done. 
\end{proof}

The theorem makes it easy to check the smoothness of simple examples
such as the toric variety of the hexagon in Example~\ref{2.hexagon} or
the rational normal scroll $S_{a,b}$ of Example~\ref{2.rnscroll}
(Exercise~\ref{2.4.easyexer}).

We also note the following useful fact, which you will prove in
Exercise~\ref{2.4.smvaexer}

\begin{proposition}
\label{2.smva}
Every smooth $n$-dimensional lattice polytope $P \subseteq \mr \simeq
\R^n$ is very ample.\qed
\end{proposition}

Here is an example of a smooth reflexive polytope whose dual is not smooth.

\begin{example}
\label{2.dualnotsmooth}
Let $P = (n+1)\Delta_n - (1,\dots,1) \subseteq \R^n$, where $\Delta_n$ is
the standard $n$ simplex.  Proposition~\ref{2.norfanunchanged} implies
that $P$ and $\Delta_n$ have the same normal fan, so that $P$ and
$\xp$ are smooth.  

The polytope $P$ has the following further interesting properties
(Exercise~\ref{2.4.dualnotsmoothexer}).  First, $P$ has the facet
presentation
\begin{align*}
x_i &\ge -1,\quad i = 1,\dots,n,\\
 -x_1 - \cdots -x_n &\ge -1,
\end{align*}
so that $P$ is reflexive with dual 
\[
P^\circ = \Conv(e_0,e_1,\dots,e_n),\quad e_0 = -e_1 - \cdots -e_n.
\]
The normal fan of $P^\circ$ consists of cones over the faces of $P$.
In particular, the cone of $\Sigma_{P^\circ}$ corresponding to the
vertex $e_0 \in P^\circ$ is the cone
\[
\sigma_{e_0} = \Conv(v_1,\dots,v_n),\quad v_i = e_0 + (n+1)e_i.
\]
Figure~10 shows $P$ and the cone $\sigma_{e_0}$ when  $n = 2$.
\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.9in]{ch2fig10.ps}
\end{center}
\caption{The cone $\sigma_{e_0}$ of the normal fan of $P^\circ$}  
\end{figure} 

For general $n$, observe that $v_i-v_j = (n+1)(e_i-e_j)$.  This makes
it easy to see that $\Z v_1 + \dots + \Z v_n$ has index $(n+1)^{n-1}$ in
$\Z^n$.  Thus $\sigma_{e_0}$ is not smooth when $n \ge 2$.  It follows
that the ``dual'' toric variety $X_{P^\circ}$ is singular for $n\ge2$.
Later we will construct $X_{P^\circ}$ as the quotient of $\P^n$ under
the action of a finite group $G \simeq (\Z/(n+1)\Z)^{n-1}$.\eoe
\end{example}

\begin{example}
\label{2.p112fan}
Consider $P = \Conv(0,2e_1,e_2) \subseteq \R^2$.  Since $P$ is very
ample, the lattice points $P\cap\Z^2 = \{0,e_1,2e_1,e_2\}$ give the
map $(\Cstar)^2 \to \P^3$ defined by
\[
(s,t) \longmapsto (1,s,s^2,t)
\]
such that $\xp$ is the Zariski closure of the image.  If $\P^3$ has
homogeneous coordinates $y_0,y_1,y_2,y_3$, then we have
\[
\xp = \V(y_0y_2 - y_1^2) \subseteq \P^3.
\]
Comparing this to Example~\ref{2.p112}, we see that $\xp$ is the
weighted projective space $\P(1,1,2)$.  Later we will learn the
systematic reason why this is true.

The variety $\xp$ is not smooth.  By working on the affine piece
$\xp\cap U_3$, one can check directly that $(0,0,0,1)$ is a singular
point of $\xp$.  

We can also use Theorem~\ref{2.smpolythm} and the normal fan of $P$,
shown in Figure~11.  One can check that the cones $\sigma_0$ and
$\sigma_1$ are smooth, but $\sigma_2$ is not,
\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.9in]{ch2fig11.ps}
\end{center}
\caption{The polygon giving $\P(1,1,2)$ and its normal fan}  
\end{figure}
so that $\Sigma_P$ is 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~\ref{2.smpolythm} implies that $\xp$ is
not smooth.

If you look carefully, you will see that $\sigma_2$ is the \emph{only}
nonsmooth cone of the normal fan $\Sigma_P$.  Once we study the
correspondence between cones and orbits in Chapter~\ref{normal}, we
will see that the non-smooth cone $\sigma_2$ corresponds to the
singular point $(0,0,0,1)$ of $\xp$.\eoe
\end{example}

\subsection*{\emph{Products of Projective Toric Varieties}} Our final
task is to understand the toric variety of a product of polytopes.
Let $P_i \subseteq (M_i)_\R \simeq \R^{n_i}$ be lattice polytopes with
$\dim P_i = n_i$ for $i = 1,2$.  This gives a lattice polytope $P_1
\times P_2 \subset (M_1\times M_2)_\R$ 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
\[
X_{P_i} \hookrightarrow \P^{s_i-1},\quad s_i = |P_i\cap M_i|,
\]
so that by Proposition~\ref{2.prodsub}, $X_{P_1}\times X_{P_2}$ is a
subvariety of $\P^{s_1-1} \times \P^{s_1-1}$.  Using the Segre
embedding 
\[
\P^{s_1-1} \times \P^{s_2-1} \hookrightarrow \P^{s-1}, \quad s = s_1s_2,
\]
we get an embedding
\begin{equation}
\label{2.xp1xp2}
X_{P_1}\times X_{P_2} \hookrightarrow \P^{s-1}.
\end{equation}
We can understand this projective variety as follows.

\begin{theorem}
\label{2.prodthm}
If $P_1$ and $P_2$ are very ample, then
\begin{thmenum}
\item $P_1 \times P_2 \subset (M_1\times M_2)_\R$ is a very ample
polytope with lattice points
\[
(P_1 \times P_2)\cap (M_1\times M_2) = (P_1 \cap M_1) \times (P_2 \cap
M_2).
\]
Thus the integer $s$ defined above is $s = |(P_1 \times P_2)\cap
(M_1\times M_2)|$.  
\item The image of the embedding $X_{P_1}\times X_{P_2}
\hookrightarrow \P^{s-1}$ coming from the very ample polytope $P_1
\times P_2$ equals the image of \eqref{2.xp1xp2}.
\item $X_{P_1\times P_2} \simeq X_{P_1}\times X_{P_2}$.
\end{thmenum}
\end{theorem}

\begin{proof}
For part (a), the assertions about lattice points are clear.  
The vertices of $P_1\times P_2$ consist of ordered pairs $(v_1,v_2)$
where $v_i$ is a vertex of $P_i$ (Exercise~\ref{2.4.prodthmexer}).
Given such a vertex, we have
\[
(P_1 \times P_2)\cap (M_1\times M_2) - (v_1,v_2) = (P_1\cap M_1 -
v_1)\times (P_2\cap M_2 - v_2).
\]
Since $P_i$ is very ample, we know that $\N(P_i\cap M_i - v_i)$ is
saturated in $M_i$.  From here, it follows easily that $P_1 \times
P_2$ 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 $T_{N_1}
\times T_{N_2}$ is Zariski dense in $X_{P_1}\times X_{P_2}$
(Exercise~\ref{2.4.prodthmexer}).  When combined with the Segre
embedding, it follows that $X_{P_1}\times X_{P_2}$ is the Zariski
closure of the image of the map
\[
T_{N_1} \times T_{N_2} \longrightarrow \P^{s_1s_2-1}
\]
given by the characters $\chi^{m}\chi^{m'}$, where $m$ ranges over the
$s_1$ elements of $P_1\cap M_1$ and $m$ ranges over the $s_2$ elements
of $P_2\cap M_2$.   When we identify $T_{N_1} \times T_{N_2}$ with
$T_{N_1\times N_2}$, the product $\chi^{m}\chi^{m'}$ becomes the
character $\chi^{(m,m')}$, so that the above map coincides with the map
\[
T_{N_\times N_2} \longrightarrow \P^{s-1}
\]
coming from the product polytope $P_1 \times P_2 \subset (M_1\times
M_2)_\R$.  Part (b) follows, and part (c) is an immediate consequence.
\end{proof}

Here is an obvious example.

\begin{example}
\label{2.prodex}
Since $\P^n$ is the toric variety of the standard $n$-simplex
$\Delta_n$, it follows that $\P^n\times\P^n$ is the toric variety of
$\Delta_n \times \Delta_m$.  

This also works for more than two factors.  Thus
$\P^1\times\P^1\times\P^1$ is the toric variety of the cube pictured
in Figure~8.
\end{example}

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~\ref{2.4.prodnfexer}).

\begin{proposition}
\label{2.prodnf}
Let $P_i \subseteq (M_i)_\R \simeq \R^{n_i}$ be lattice polytopes with
$\dim P_i = n_i$ for $i = 1,2$.  Then
\[
\hspace*{1.68truein}\Sigma_{P_1\times P_2} = 
\Sigma_{P_1}\times \Sigma_{P_2}.\hspace*{1.68truein} \qed
\]
\end{proposition}

Here is an easy example.

\begin{example}
\label{2.P1P1}
The normal fan of an interval $[a,b] \subset \R$, where $a < b$ in $\Z$, is
given by
\[
\begin{picture}(220,25)
\thicklines
\put(10,15){\line(1,0){200}}
\put(110,15){\circle*{4}}
\put(108,4){$0$}
\put(60,5){$\sigma_1$}
\put(160,5){$\sigma_0$}
\end{picture}
\]
The corresponding toric variety is $\P^1$.  The cartesian product of
two such intervals is a lattice rectangle whose toric variety is $\P^1
\times \P^1$ by Theorem~\ref{2.prodthm}.  If we set $\sigma_{ij} =
\sigma_i\times\sigma_j$, then Proposition~\ref{2.prodnf} gives the
normal fan given in Figure~12.
\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.65in]{ch2fig12.ps}
\end{center}
\caption{The normal fan of a lattice rectangle giving $\P^1 \times \P^1$ }  
\end{figure}

We will revisit this example in Chapter~\ref{normal} when we construct
toric varieties directly from fans. \eoe 
\end{example}

Proposition~\ref{2.prodnf} suggests a different way to think about the
product.  Let $v_i$ range over the vertices of $P_i$.  Then the
$\sigma_{v_i}$ are the maximal cones in the normal fan $\Sigma_{P_i}$,
which implies that
\begin{equation}
\label{2.xpicover}
X_{P_i} = {\textstyle\bigcup_{v_i}} U_{\sigma_{v_i}}.
\end{equation}
Thus
\begin{align*}
X_{P_1} \times X_{P_2}\  &= \
\big({\textstyle\bigcup_{v_1}} U_{\sigma_{v_1}}\big) \times 
\big({\textstyle\bigcup_{v_2}} U_{\sigma_{v_2}}\big) \\
\ &= \ \ {\textstyle\bigcup_{(v_1,v_2)}} U_{\sigma_{v_1}} \!
\times U_{\sigma_{v_2}}\\
\ &= \ \ {\textstyle\bigcup_{(v_1,v_2)}} U_{\sigma_{v_1} \!
\times \sigma_{v_2}}\\
\ &= \ \ {\textstyle\bigcup_{(v_1,v_2)}} U_{\sigma_{(v_1,v_2)}} = X_{P_1
    \times P_2}.
\end{align*}
In this sequence of equalities, the first follows from
\eqref{2.xpicover}, the second is obvious, the third uses
Exercise~\ref{1.3.prod} of Chapter~\ref{affine}, the fourth uses
Proposition~\ref{2.prodnf}, and the last follows since $(v_1,v_2)$
ranges over all vertices of $P_1 \times P_2$.

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~\ref{affine}.  We will
use this idea in Chapter~\ref{normal} to define the cartesian product
of abstract varieties.

\exercises

\small

\begin{exercise}
\label{2.4.easyexer}
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.
\end{exercise}

\begin{exercise}
\label{2.4.smvaexer}
Prove Proposition~\ref{2.smva}.
\end{exercise}

\begin{exercise}
\label{2.4.dualnotsmoothexer}
Consider the polytope $P = (n+1)\Delta_n - (1,\dots,1)$ from
Example~\ref{2.dualnotsmooth}. 
\begin{thmenum}
\item Verify the facet presentation of $P$ given in the example.
\item What is the facet presentation of $P^\circ$?  Hint:\ You know
the vertices of $P$.
\item Let $v_i = e_0 + (n+1)e_i$, where $i = 1,\dots,n$ and $e_0 =
  -e_1 - \cdots - e_n$, and let $L = \Z v_1 + \cdots + \Z v_n$.  Use
  the hint given in the text to prove $\Z^n/L \simeq
  (\Z/(n+1)\Z)^{n-1}$.  This shows that the index of $L$ in $\Z^n$ is
  $(n+1)^{n_1}$, as claimed in the text.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{2.4.prodthmexer}
Let $P_i \subseteq (M_i)_\R \simeq \R^{n_i}$ be lattice polytopes with
$\dim P_i = n_i$ for $i = 1,2$.  Also let $S_i$ be the set of vertices
of $P_i$.
\begin{thmenum}
\item Use supporting hyperplanes to prove that every element of $S_1
  \times S_2$ is a vertex of $P_1 \times P_2$.
\item Prove that $P_1\times P_2 = \Conv(S_1\times S_2)$ and conclude
  that $S_1 \times S_2$ is the set of vertices of $P_1 \times P_2$.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{2.4.prodnfexer}
The goal of this exercise is to prove Proposition~\ref{2.prodnf}.  We
know from Exercise~\ref{2.4.prodthmexer} that the vertices of
$P_1\times P_2$ are the ordered pairs $(v_1,v_2)$ where $v_i$ is a
vertex of $P_i$.
\begin{thmenum}
\item Adapt the argument of part (a) of Theorem~\ref{2.prodthm} to
  show that $C_{(v_1,v_2)} = C_{v_1} \times C_{v_2}$.  Taking duals, we
  see that the maximal cones of $\Sigma_{P_1\times P_2}$ are
  $\sigma_{(v_1,v_2)} = \sigma_{v_1} \times \sigma_{v_2}$.
\item Given rational polyhedral cones $\sigma_i \subseteq (N_i)_\R$ and
  faces $\tau_i \subseteq \sigma_i$, prove that $\tau_1 \times \tau_2$
  is a face of $\sigma_1\times\sigma_2$ and that all faces of
  $\sigma_1\times\sigma_2$ arise this way.
\item Prove that $\Sigma_{P_1\times P_2} = \Sigma_{P_1}\times
  \Sigma_{P_2}$. 
\end{thmenum}
\end{exercise}

% exercises not mentioned in text go here

\begin{exercise}
\label{2.3.reflexiveexer1}
Consider positive integers $1 = q_0 \le q_2 \le \cdots \le q_n$ with
the property that $q_i | \sum_{i=0}^n q_i$ for $i = 0,\dots,n$.  Set
$k_i = \big( \sum_{i=0}^n q_i\big)/q_i$ for $i = 1,\dots,n$ and let
\[
P_{q_0,\dots,q_n} = \Conv(0,k_1e_1,k_2e_2,\dots,k_n e_n) - (1,\dots,1).
\]
Prove that $P_{q_0,\dots,q_n}$ is reflexive and explain how it relates
to Example~\ref{2.dualnotsmooth}.  We will prove later that the toric
variety of this polytope is the weighted projective space
$\P(q_0,\dots,q_n)$.  
\end{exercise}

\begin{exercise}
\label{2.3.reflexiveexer2}
The \emph{Sylvester sequence} is defined by $a_0 = 2$ and $a_{k+1} = 1
+ a_1 a_2 \cdots a_k$.  It begins $2,3,7,43,1807,\dots$ and is
described in \cite[A000058]{sloane}.  Now fix a positive integer $n
\ge 3$ and define $q_0,\dots,q_n$ by $q_0 = q_1 = 1$ and $q_i =
2(a_{n-1} - 1)/a_{n-i}$ for $i = 2,\dots,n$.  For $n = 3$ and $4$ this
gives $1,1,4,6$ and $1,1,12,28,42$.  Prove that $q_0,\dots,q_n$
satisfies the conditions of Exercise~\ref{2.3.reflexiveexer1} and
hence gives a reflexive simplex, denoted $S_{Q'_n}$ in \cite{nill}.
This paper proves when $n \ge 4$, $S_{Q'_n}$ has the largest volume of
all $n$-dimensional reflexive simplices and conjectures that it also
has the largest number of lattice points.
\end{exercise}




\normalsize



\endinput


\section{Polyhedra and Toric Varieties}

Unified treatment of affine and projective toric varieties using
polyhedra.  

general facts: poly


\bigskip

{\bf stuff from Hal}

\bigskip

Section 1:  {$X_A \subseteq \P^{\,n}$ (lattice pts in a hyperplane), RNC,
  rational quartic, homogeneous toric ideal}

\bigskip

Section2: {$X_{P \cap M} \subseteq \P^{\,n}$}
 
\begin{itemize}
    \item{lattice polytope}
    \item{normal fan}
    \item{multiples of $P$}
    \item{affine opens, gluing, duals}
    \item{complete fan}
    \item{unimodular, simplicial, reflexive polytopes}
    \item{$P=Q+R$ implies $X_P \longrightarrow X_R$}
    \item{fan over faces $P$ vs. normal fan of $P$}
    \item{examples include simplices, products, rational normal
    scroll (with determinantal presentation), $\P(1,1,2)$}
\end{itemize}

\bigskip

Section 3: {Polyhedra}

\begin{itemize}
    \item{$P=Q+C$ polytope plus cone}
    \item{$X_{mP} \longrightarrow X_C \times P^n \longrightarrow X_C$}
    \item{Normal fan $\Sigma_P$, $|\Sigma_P|^\vee = C$}
    \item{why $mP$ gives embedding for $m\gg 0$ (for minimal
       generators of cone, m = lcm(height mingens), $mP$ mingens of
       height one)}
    \item{example---truncated Egyptian pyramid, blowup of $\C^2$, $\C^3$}
\end{itemize} 
