\chapter{Affine Toric Varieties}
\label{affine}

\setcounter{section}{-1}

\section{Background:\ Affine Varieties}

We begin with the algebraic geometry needed for our study of affine
toric varieties.  Our discussion assumes Chapters 1--5 and 9 of
\cite{iva}.

\subsection*{\emph{Coordinate Rings}} An ideal $I \subseteq S =
\C[x_0,\dots,x_n]$ gives an affine variety
\[
\V(I) = \{p \in \C^n \mid f(p) = 0 \text{ for all } f \in I\}
\]
and an affine variety $V \subseteq \C^n$ gives the ideal
\[
\I(V) = \{f \in S \mid f(p) = 0 \text{ for all } p \in V\}.
\]
By the Hilbert Basis Theorem, an affine variety $V$ is defined by the
vanishing of finitely many polynomials in $S$, and for any ideal $I$,
the Nullstellensatz tells us that $\I(\V(I)) = \sqrt{I} = \{f \in S
\mid f^{k} \in I \text{ for some } k \ge 1\}$ since $\C$ is
algebraically closed.  The most important algebraic object
associated to $V$ is its \emph{coordinate ring}
\[
\C[V] = S/\I(V).
\]
Elements of $\C[V]$ can be interpreted as the $\C$-valued polynomial
functions on $V$.  Note that $\C[V]$ is a $\C$-algebra, meaning that
its vector space structure is compatible with its ring structure.
Here are some basic facts about coordinate rings:
\begin{coxitem}
\item $\C[V]$ is an integral domain $\Leftrightarrow$ $\I(V)$ is a
prime ideal $\Leftrightarrow$ $V$ is irreducible.
\item Polynomial maps (also called \emph{morphisms}) $F : V_1 \to V_2$
between affine varieties correspond to $\C$-algebra homomorphisms $F^*
: \C[V_2] \to \C[V_1]$, where $F^*(g) = g\circ F$ for $g \in \C[V_2]$.
\item Two affine varieties are isomorphic if and only if their
coordinate rings are isomorphic $\C$-algebras.
\item A point $p$ of an affine variety $V$ gives the maximal ideal
\[
\{f \in \C[V] \mid f(p) = 0\} \subseteq \C[V],
\]
and all maximal ideals of $\C[V]$ arise this way.
\end{coxitem}
Coordinate rings of affine varieties can be characterized as follows
(Exercise~\ref{1.0.1}).

\begin{lemma}
\label{1.charcoord}
A $\C$-algebra $R$ is isomorphic to the
coordinate ring of an affine variety if and only if $R$ is a finitely
generated $\C$-algebra with no nonzero nilpotents, i.e., if $f \in R$
satisfies $f^{m} = 0$, then $f = 0$.\qed
\end{lemma}

To emphasize the close relation between $V$ and $\C[V]$, we 
sometimes write
\begin{equation}
\label{0.specdef}
V = \Spec(\C[V]).
\end{equation}
This can be made canonical by identifying $V$ with the set of maximal
ideals of $\C[V]$ via the fourth bullet above.  More generally, one
can take any commutative ring $R$ and define the \emph{affine scheme}
$\Spec(R)$.  The general definition of Spec uses all prime ideals of
$R$, not just the maximal ideals as we have done.  Thus some authors
would write \eqref{0.specdef} as $V = \mathrm{Specm}(\C[V])$, the
maximal spectrum of $\C[V]$).  Readers wishing to learn more about
affine schemes should consult \cite{eh} and \cite{hartshorne}.

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

Given a subset $S \subseteq V$, its closure $\overline{S}$ in the
Zariski topology is the smallest subvariety of $V$ containing $S$.  We
call $\overline{S}$ the \emph{Zariski closure} of $S$.  It is easy to
give examples where this differs from the closure in the classical
topology.
 
\subsection*{\emph{Affine Open Subsets and Localization}} Some Zariski
open subsets of an affine variety $V$ are themselves affine varieties.
Given $f \in \C[V] \setminus \{0\}$, let 
\[
V_f = \{p \in V \mid f(p) \ne 0\} \subseteq V.
\]
Then $V_f$ is Zariski open in $V$ and is also an affine variety, as we
now explain.

Let $V \subseteq \C^n$ have $\I(V) = \langle f_1,\dots,f_s\rangle$ and
pick $g \in \C[x_1,\ldots,x_n]$ representing $f$.  Then $V_f = V -
\V(g)$ is Zariski open in $V$.  Now consider a new variable $y$ and
let $W = {\bf V}(f_1,\ldots,f_s,1-gy) \subseteq \C^n\times\C$.  Since
the projection map $\C^n\times\C \to \C^n$ maps $W$ bijectively onto
$V_f$, we can identify $V_f$ with the affine variety $W \subseteq
\C^n\times\C$.

When $V$ is irreducible, the coordinate ring of $V_f$ is easy to
describe.  Let $\C(V)$ be the field of fractions of the integral
domain $\C[V]$.  Recall that elements of $\C(V)$ give rational
functions on $V$.  Then let
\begin{equation}
\label{1.localize}
\C[V]_f = \{g/f^{\ell} \in \C(V) \mid g \in \C[V],\ \ell \ge 0\}.
\end{equation}
In Exercise~\ref{1.0.2} you will prove that $\Spec(\C[V]_f)$ is the
affine variety $V_f$.

\begin{example}
\label{1.torus}
The $n$-dimensional torus is the affine open subset 
\[
\torus = \C^n \setminus \V(x_1\cdots x_n) \subseteq \C^n,
\]
with coordinate ring
\[
\C[x_0,\dots,x_n]_{x_1\cdots x_n} = \C[x_1^{\pm1},\dots,x_n^{\pm1}].
\]
Elements of this ring are called \emph{Laurent polynomials}. \eoe
\end{example}

The ring $\C[V]_f$ from \eqref{1.localize} is an example of
\emph{localization}.  In Exercise~\ref{1.0.2} you will show how to
construct this ring for all affine varieties, not just irreducible
ones.  The general concept of localization is discussed in standard
texts in commutative algebra such as \cite[Chapter~3]{am} and
\cite[Chapter~2]{eisenbud}.

\subsection*{\emph{Normal Affine Varieties}} Let $R$ be an integral
domain with field of fractions $K$.  Then $R$ is \emph{normal} if
every element of $K$ which is integral over $R$ (meaning that it is a
root of a monic polynomial in $R[x]$) actually lies in $R$.  For
example, any UFD is normal (Exercise~\ref{1.0.3}).

\begin{definition}
\label{1.normaldef}
An irreducible affine variety $V$ is \emph{\bfseries normal} if its
coordinate ring $\C[V]$ is normal.
\end{definition}

For example, $\C^n$ is normal since its coordinate ring
$\C[x_1,\dots,x_n]$ is a UFD and hence normal.  Here is an
example of a non-normal affine variety.

\begin{example}
\label{1.cuspnotnormal}
Let $C = {\bf V}(x^{3}-y^{2}) \subseteq \C^2$.  This is an irreducible
plane curve with a cusp at the origin.  It is easy to see that $\C[C]
= \C[x,y]/\langle x^{3}-y^{2}\rangle$.  Now let $\bar{x}$ and
$\bar{y}$ be the cosets of $x$ and $y$ in $\C[C]$ respectively.  This
gives $\bar{y}/\bar{x} \in \C(C)$.  A computation shows that
$\bar{y}/\bar{x} \notin \C[C]$ and that $(\bar{y}/\bar{x})^2 =
\bar{x}$.  Consequently $\C[C]$ and hence $C$ are not normal.

We will see below that $C$ is an affine toric variety. \eoe
\end{example}

An irreducible affine variety $V$ has a \emph{normalization} defined
as follows.  Let
\[
\C[V]' = \{\alpha \in \C(V): \alpha\ \text{is integral over}\
\C[V]\}.
\]
We call $\C[V]'$ the {\it integral closure} of $\C[V]$.  One can show
that $\C[V]'$ is normal and (with more work) finitely generated as a
$\C$-algebra (see \cite[Corollary~13.13]{eisenbud}).  This gives the
normal affine variety
\[
V' = {\rm Spec}(\C[V]')
\]
We call $V'$ the {\it normalization\/} of $V$.  The natural inclusion
$\C[V] \subseteq \C[V]' = \C[V']$ corresponds to a map $V' \to V$.  This
is the {\it normalization map\/}.

\begin{example}
\label{1.normalizecusp}
We saw in Example~\ref{1.cuspnotnormal} that the curve $C \subseteq
\C^2$ defined by $x^{3} = y^{2}$ has elements $X,Y \in \C[C]$ such
that $Y/X \notin \C[C]$ is integral over $\C[C]$.  In
Exercise~\ref{1.0.4} you will show that $\C[\bar{y}/\bar{x}] \subseteq
\C(C)$ is the integral closure of $\C[C]$ and that the normalization
map is the map $\C \to C$ defined by $t \mapsto (t^2,t^3)$. \eoe
\end{example}

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

In Exercise~\ref{1.0.5} you will prove some properties of normal
domains that will be used in \S3 when we study normal affine toric
varieties.

\subsection*{\emph{Smooth Points of Affine Varieties}} In order to
define a smooth point of an affine variety $V$, we first need to
define \emph{local rings} and \emph{Zariski tangent spaces}.  For
simplicity, we assume that $V$ is irreducible.  The \emph{local ring}
of $V$ at $p$ is
\[
\cO_{V,p} = \{f/g \in \C(V) \mid f,g \in \C[V] \text{ and } g(p) \ne
0\}. 
\]
Thus $\cO_{V,p}$ consists of all rational functions on $V$ that are
defined at $p$.  Inside of $\cO_{V,p}$ we have the maximal ideal
\[
\mf_{V,p} = \{\varphi \in \cO_{V,p} \mid \varphi(p) = 0\}. 
\]
One can prove without difficulty that $\mf_{V,p}$ is the unique
maximal ideal of $\cO_{V,p}$.  Rings with this property are called
\emph{local rings}.

The \emph{Zariski tangent space} of $V$ at $p$ is defined to be
\[
T_p(V) = \mathrm{Hom}_\C(\mf_{V,p}/\mf_{V,p}^2,\C).
\]
In Exercise~\ref{1.0.6} you will verify that $\dim T_p(\C^n) = n$ for
every $p \in \C^n$.  According to \cite[p.\ 32]{hartshorne}, we can
compute the Zariski tangent space of a point in an affine variety as
follows.

\begin{lemma}
\label{1.computeZar}
Let $V \subseteq \C^n$ be a affine variety and let $p \in V$.  Also
assume that $\I(V) = \langle f_1,\ldots,f_s\rangle \subseteq
\C[x_1,\dots,x_n]$.  For each $i$, let
\[
d_p(f_i) = \frac{\partial f_i}{\partial x_1}(p)\, x_1 + \cdots +
\frac{\partial f_i}{\partial x_n}(p)\, x_n.
\]
The the Zariski tangent $T_p(V)$ is isomorphic to the subspace of
$\C^n$ defined by the equations $d_p(f_1) = \cdots = d_p(f_s) =
0$.  In particular, $\dim T_p(V) \le n$.  \qed 
\end{lemma}

\begin{definition}
\label{1.smoothdef}
A point $p$ of an affine variety $V$ is \emph{\bfseries smooth} or
\emph{\bfseries nonsingular} if $\dim T_p(V) = \dim V$.  The point
$p$ is \emph{\bfseries singular} if it is not smooth.  Finally, $V$ is
\emph{\bfseries smooth} if every point of $V$ is smooth.
\end{definition}

Since $\dim T_p(\C^n) = n$ for every $p \in \C^n$, we see that $\C^n$
is smooth.  For an irreducible affine variety $V \subseteq \C^n$ of
dimension $d$, fix $p \in V$ and write $\I(V) = \langle f_1,\ldots,
f_s\rangle$.  Using Lemma~\ref{1.computeZar}, it is straightforward to
show that $V$ is smooth at $p$ if and only if the Jacobian matrix
\begin{equation}
\label{1.jacobian}
J_p(f_1,\ldots,f_s) = \Big(\frac{\partial f_i}{\partial
x_j}(p)\Big)_{1\le i \le s,1\le j\le n}
\end{equation}
has rank $n-d$ (Exercise~\ref{1.0.7}).  Here is a simple example.

\begin{example}
\label{1.cuspnotsmooth}
As noted in Example~\ref{1.cuspnotnormal}, the plane curve $C$ defined
by $x^{3} = y^{2}$ has $\I(C) = \langle x^{3} - y^{2}\rangle
\subseteq \C[x,y]$.  A point $p = (a,b) \in C$ has Jacobian
\[
J_p = (3a^2,2b),
\]
so the origin is the only singular point of $C$.
\end{example}

Since $T_p(V) = {\rm Hom}_\C(\mf_{V,p}/\mf_{V,p}^2,\C)$, we see that
$V$ is smooth at $p$ when $\dim V$ equals the dimension of
$\mf_{V,p}/\mf_{V,p}^2$ as a vector space over $\cO_{V,p}/\mf_{V,p}$.
In terms of commutative algebra, this means that $p \in V$ is smooth
if and only if $\cO_{V,p}$ is a \emph{regular local ring}.  See
\cite[p.~123]{am} or \cite[10.3]{eisenbud}.  

We can relate smoothness and normality as follows.

\begin{proposition}
\label{1.smoothnor}
A smooth irreducible affine variety $V$ is normal.
\end{proposition}

\begin{proof}
As noted above, the smoothness of $V$ implies that $\cO_{V,p}$ is a
regular local ring for all $p \in V$.  A powerful result of
commutative algebra states that a regular local ring is a UFD (see
\cite[Theorem~19.19]{eisenbud}).  Hence $\cO_{V,p}$ is normal for all
$p$.  In \S0 of Chapter~\ref{normal} we will see that
\[
\C[V] = \bigcap_{p \in V} \cO_{V,p},
\]
which by Exercise~\ref{1.0.5} implies that $\C[V]$ is normal.
\end{proof}

The converse of Propostion~\ref{1.smoothnor} can fail.  We will see in
\S3 that the affine variety $\V(xy-zw) \subseteq \C^4$ is normal,
yet $\V(xy-zw)$ is singular at the origin.

\subsection*{\emph{Products of Affine Varieties}} Given affine
varieties $V_1$ and $V_2$, there are several ways to show that the
cartesian product $V_1\times V_2$ is an affine variety.  The most
direct way is to proceed as follows.  Let $V_1\subseteq \C^m =
\Spec(\C[x_1,\ldots,x_m])$ and $V_2 \subseteq \C^n =
\Spec(\C[y_1,\ldots,y_n])$.  Take $\I(V_1) = \langle f_1, \ldots,
f_s\rangle$ and $\I(V_2) = \langle g_1, \ldots, g_t\rangle$.  Since
the $f_i$ and $g_j$ depend on separate sets of variables, it follows
that
\[
V_1\times V_2 =\V(f_1,\ldots,f_s,g_1,\ldots,g_t) \subseteq \C^{m+n}
\]
is an affine variety.

A fancier method is to use the mapping properties of the product.
This will also give an instrinic description of its coordinate ring.
Given $V_1$ and $V_2$ as above, $V_1\times V_2$ should be an affine
variety with projections $\pi_i : V_1\times V_2 \to V_i$ such that
whenever we have a diagram 
\[
\SelectTips {cm}{}
\xymatrix{
W \ar@/^1pc/[drr]^{\phi_1} \ar@/_1pc/[ddr]_{\phi_2} \ar@{.>}[dr]|{\nu} & & \\
& V_1 \times V_2 \ar[r]_(.6){\pi_1} \ar[d]^{\pi_2} & V_1\\
& V_2 &
}
\]
where $\phi_i : W \to V_i$ are morphisms from an affine variety $W$,
there should be a unique morphism $\nu$ (the dotted arrow) that makes
the diagram commute, i.e., $\pi_i\circ\nu = \phi_i$.  For the
coordinate rings, this means that whenever we have a diagram
\[
\SelectTips {cm}{}
\xymatrix{
& \C[V_2] \ar[d]_{\pi_2^*} \ar@/^1pc/[ddr]^{\phi^*_2} & \\
\C[V_1] \ar[r]^(.45){\pi_1^*} \ar@/_1pc/[drr]_{\phi^*_1} & \C[V_1\times V_2]
\ar@{.>}[dr]|{\nu^*} \\
& & \C[W]
}
\]  
with $\C$-algebra homomorphisms $\phi_i^*: \C[V_i] \to \C[W]$, there
should be a unique $\C$-algebra homomorphism $\nu^*$ (the dotted
arrow) that makes the diagram commute.  By the universal mapping
property of the \emph{tensor product} of $\C$-algebras, there is
exactly one algebra that does this (up to isomorphism), namely
$\C[V_1] \otimes_\C \C[V_2]$.  This is a finitely generated
$\C$-algebra with no nilpotents, so it is the coordinate ring of an
affine variety by Lemma~\ref{1.charcoord}.  We could take this as the
definition of $V_1\times V_2$.
 
\begin{example}
\label{1.firstproduct}
Let $V$ be an affine variety.  Since $\C^n =
\Spec(\C[y_1,\dots,y_n])$, the product $V\times\C^n$ has coordinate
ring
\[
\C[V] \otimes_\C \C[y_1,\dots,y_n] = \C[V][y_1,\dots,y_n].
\]
If $V$ is contained in $\C^m$ with $\I(V) = \langle f_1,\dots,f_s
\rangle \subseteq \C[x_1,\dots,x_m]$, it follows that
\[
\I(V\times\C^n) = \langle f_1,\dots,f_s \rangle \subseteq
\C[x_1,\dots,x_m,y_1,\dots,y_n].
\]
For later purposes, we also note that the coordinate ring of $V \times
\torus$ is
\[
\hspace*{1.1truein}\C[V] \otimes_\C \C[y_1^{\pm1},\dots,y_n^{\pm1}] =
\C[V][y_1^{\pm1},\dots,y_n^{\pm1}].\hspace*{57.33truept}\text{\eoe}
\]
\end{example}

Given affine varieties $V_1$ and $V_2$, we note that the Zariski
topology on $V_1\times V_2$ is usually \emph{not} the product of the
Zariski topologies on $V_1$ and $V_2$.

\begin{example}
\label{1.prodbad}
Consider $\C^2 = \C \times \C$.  By definition, a basis for the
product of the Zariski topologies consists of sets $U_1 \times U_2$
where $U_i$ are Zariski open in $\C$.  Such a set is the complement of
a union of collections of ``horizontal'' and ``vertical'' lines in
$\C^2$.  This makes it easy to see that Zariski closed sets in $\C^2$
such as $\V(y - x^{2})$ cannot be closed in the product
topology.\eoe 
\end{example}
 
\exercises

\small

\begin{exercise}
\label{1.0.1}
Prove Lemma~\ref{1.charcoord}. Hint:\ You will need the
Nullstellensatz. 
\end{exercise}

\begin{exercise}
\label{1.0.2}
Let $R$ be a finitely generated $\C$-algebra without nilpotents
as in Lemma~\ref{1.charcoord} and let $f \in R$ be nonzero.  The
\emph{localization} $R_f$ consists of all formal expressions
$g/f^{k}$, $g \in R$, modulo the equivalence relation
\[
g/f^{k} \sim h/f^{\ell} \iff f^{s}(f^{\ell} g - f^{k} h) = 0
\text{ for some } s \ge 0.
\]
\begin{thmenum}
\item Show that $R_f$ satisfies $\Spec(R_f) = \Spec(R)_f$.
\item Show that $R_f$ is given by \eqref{1.localize} when $R$ is an
integral domain. 
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{1.0.3}
Prove that a UFD is normal.
\end{exercise}

\begin{exercise}
\label{1.0.4}
In the setting of Example~\ref{1.normalizecusp}, show that
$\C[\bar{y}/\bar{x}] \subseteq \C(C)$ is the integral closure of
$\C[C]$ and that the normalization $\C \to C$ is defined by $t \mapsto
(t^2,t^3)$.
\end{exercise}

\begin{exercise}
\label{1.0.5}
In this exercise, you will prove some properties of normal
domains need for \S3.
\begin{thmenum}
\item Let $R$ be a normal domain with field of fractions $K$ and fix
  $f \in R$.  Prove that the localization $R_f$ is normal.
\item Let $R_\alpha$. $\alpha \in A$, be normal domains with the same
  field of fractions $K$.  Prove that the intersection
  $\bigcap_{\alpha\in A} R_\alpha$ is normal.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{1.0.6}
Prove that $\dim T_p(\C^n) = n$ for all $p \in \C^n$.
\end{exercise}

\begin{exercise}
\label{1.0.7}
Use Lemma~\ref{1.computeZar} to prove the claim made in the text that
smoothness is determined by the rank of the Jacobian matrix
\eqref{1.jacobian}.
\end{exercise}

\normalsize

\section{Introduction to Affine Toric Varieties}

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

\subsection*{\emph{The Torus}} The affine variety $\torus$ is a group
under component-wise multiplication.  A \emph{torus} $T$ is an affine
variety isomorphic to $\torus$, where $T$ inherits a group structure
from the isomorphism.  Associated to $T$ are its \emph{characters} and
\emph{one-parameter subgroups}.  We discuss each of these
briefly.

A \emph{character} of a torus $T$ is a morphism $\chi : T \to \C^*$
that is a group homomorphism.  For example, $m = (a_1,\dots,a_n)
\in \Z^n$ gives a character $\chi^m : \torus \to \C^*$ defined by
\begin{equation}
\label{1.laurentmon}
\chi^m(t_1,\dots,t_n) = t_1^{a_1} \cdots t_n^{a_n}.
\end{equation}
One can show that \emph{all} characters of $\torus$ arise this way (see
\cite[\S16]{humphreys}).  Thus the characters of $\torus$ form a group
isomorphic to $\Z^n$.

For an arbitrary torus $T$, its characters form a free abelian group
$M$ of rank equal to the dimension of $T$.  It is customary to say
that $m \in M$ gives the character $\chi^m : T \to \C^*$. 

We will need the following results concerning tori (see
\cite[\S16]{humphreys} for proofs).

\begin{proposition}
\label{1.toriprops}\
\begin{thmenum}
\item Let $T_1$ and $T_2$ be tori and let $\Phi : T_1 \to T_2$ be a
  morphism that is a group homomorphism.  Then the
  image of $\Phi$ is a torus and is closed in $T_2$.
\item Let $T$ be a torus and let $H \subseteq T$ be an irreducible
  subvariety of $T$ that is a subgroup.  Then $H$ is a torus. \qed
\end{thmenum}
\end{proposition}

Now assume that a torus $T$ acts linearly on a finite dimensional vector
space $W$ over $\C$, where the action of $t \in T$ on $w \in W$ is
denoted $t\cdot w$.  Given $m \in M$, we get the \emph{eigenspace}
\[
W_m = \{w \in W \mid t\cdot w = \chi^m(t) w \text{ for all } t \in
T\}.
\]
If $W_m \ne \{0\}$, then every $w \in W_m \setminus \{0\}$ is a
simultaneous eigenvector for all $t \in T$, with eigenvalue given by
$\chi^m(t)$.  

\begin{proposition}
\label{1.torusaction}
In the above situation, we have $W = \bigoplus_{m \in M} W_m$.\qed
\end{proposition}

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

A \emph{one-parameter subgroup} of a torus $T$ is a morphism $\chi :
\C^* \to T$ that is a group homomorphism.  For example, $u =
(b_1,\dots,b_n) \in \Z^n$ gives a one-parameter subgroup $\lambda^u :
\C^* \to \torus$ defined~by
\begin{equation}
\label{1.moncurve}
\lambda^u(t) = (t^{b_1}, \dots, t^{b_n}).
\end{equation}
All one-parameter subgroups of $\torus$ arise this way (see
\cite[\S16]{humphreys}).  It follows that the group of one-parameter
subgroups of $\torus$ is naturally isomorphic to $\Z^n$.  For an
arbitrary torus $T$, the one-parameter subgroups form a free abelian
group $N$ of rank equal to the dimension of $T$.  As with the
character group, an element $u \in N$ gives the one-parameter
subgroup $\lambda^u : \Cstar \to T$.

There is a natural bilinear pairing $\langle \ ,\ \rangle : M \times N
\to \Z$ defined as follows.
\begin{coxitem}
\item (Intrinsic) Given a character $\chi^m$ and a one-parameter
  subgroup $\lambda^u$, the composition $\chi^m \circ \lambda^u : \C^*
  \to \C^*$ is character of $\C^*$, which is given by $t \mapsto
  t^\ell$ for some $\ell \in \Z$.  Then $\langle m,u\rangle = \ell$.
\item (Concrete) If $T = \torus$ with $m = (a_1,\dots,a_n) \in \Z^n$,
  $u = (b_1,\dots,b_n) \in \Z^n$, then one computes that
\begin{equation}
\label{1.dotprod}
\langle m,u\rangle = \sum_{i=1}^n a_ib_i,
\end{equation}
i.e., the pairing is the usual dot product.  
\end{coxitem}

It follows that the characters and one-parameter subgroups of a torus
$T$ form free abelian groups $M$ and $N$ of finite rank with a pairing
$\langle \ ,\ \rangle : M \times N \to \Z$ that identifies $N$ with
$\mathrm{Hom}_\Z(M,\Z)$ and $M$ with $\mathrm{Hom}_\Z(N,\Z)$.  In terms
of tensor products, one obtains a canonical isomorphism $N \otimes_\Z
\C^* \simeq T$ via $u\otimes t \mapsto \lambda^u(t)$.  Hence it is
customary to write a torus as $T_N$.

{}From this point of view, picking an isomorphism $T_N \simeq \torus$
induces dual bases of $M$ and $N$, i.e., isomorphisms $M \simeq \Z^n$
and $N \simeq \Z^n$ that turn characters into Laurent monomials
\eqref{1.laurentmon}, one-parameter subgroups into monomial curves
\eqref{1.moncurve}, and the pairing into dot product
\eqref{1.dotprod}.

\subsection*{\emph{The Definition of Affine Toric Variety}} We now
define the main object of study of this chapter.

\begin{definition}
\label{1.afftordef}
An \emph{\bfseries affine toric variety} is an irreducible affine
variety $V$ containing a torus $T_N \simeq \torus$ as a Zariski open
subset such that the action of $T_N$ on itself extends to an action of
$T_N$ on $V$.
\end{definition}

Obvious examples of affine toric varieties are $\torus$ and $\C^n$.
Here are some less trivial examples.

\begin{example}
\label{1.cuspex}
The plane curve $C = \V(x^{3}-y^{2}) \subseteq \C^2$ has a cusp
at the origin.  This is an affine toric variety with torus
\[
C - \{0\} = C \cap (\C^*)^2 = \{(t^2,t^3) \mid t \in \C^*\} \simeq
\C^*,
\]
where the isomorphism is $(t^2,t^3) \mapsfrom t$.
Example~\ref{1.cuspnotnormal} shows that $C$ is a non-normal toric
variety. \eoe
\end{example}

\begin{example}
\label{1.xyzw}
The variety $V = \V(xy - zw) \subseteq \C^4$ is a toric variety with
torus
\[
V \cap (\C^*)^4 = \{(t_1,t_2,t_3,t_1t_2t_3^{-1}) \mid t_i \in \C^*\}
\simeq (\C^*)^3,
\]
where the isomorphism is $(t_1,t_2,t_3,t_1t_2t_3^{-1}) \mapsfrom
(t_1,t_2,t_3)$.  We will see later that $V$ is normal. \eoe
\end{example}

\begin{example}
\label{1.rnccone}
Consider the surface in $\C^{d+1}$ parametrized by the map
\[
\Phi: \C^2 \longrightarrow \C^{d+1}
\]
defined by $(s,t) \mapsto (s^d,s^{d-1}t,\dots,st^{d-1},t^d)$.  Thus
$\Phi$ is defined using all degree $d$ monomials in $s,t$.  

Let the coordinates of $\C^{d+1}$ be $x_0,\dots,x_d$ and let $I
\subseteq \C[x_0,\dots,x_d]$ be the ideal generated by the $2\times 2$
minors of the matrix
\[
\begin{pmatrix} x_0 & x_1 & \cdots & x_{d-2} & x_{d-1} \\
x_1 & x_2 & \cdots & x_{d-1} & x_{d}
\end{pmatrix},
\]
so $I = \langle x_i x_{j+1} - x_{i+1} x_j \mid 0 \le i < j \le
d-1\rangle$.  In Exercise~\ref{1.1.1} you will verify that $\Phi(\C^2)
= \V(I)$, so that $\widehat{C}_d = \Phi(\C^2)$ is an affine variety,
and that $\I(\widehat{C}_d) = I$, so that $I$ is the ideal of all
polynomials vanishing on $\widehat{C}_d$.  It follows that $I$ is
prime since $V$ is irreducible by Proposition~\ref{1.lattpt} below.
The affine surface $\widehat{C}_d$ is called the \emph{rational normal
cone of degree $d$} and is an example of a \emph{determinantal
variety}.  We will see below that $I$ is an example of a toric ideal.

It is straightforward to show that $\widehat{C}_d$ is a toric variety
with torus
\[
\Phi((\C^*)^2) = \widehat{C}_d \cap (\C^*)^{d+1} \simeq (\C^*)^2.
\]
We will study this example from the projective point of view in
Chapter~\ref{poly}. \eoe
\end{example}

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

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

Given a torus $T_N$ with character lattice $M$, a set $\cA =
\{m_1,\dots,m_s\} \subseteq M$ gives characters $\chi^{m_i} : T_N \to
\C^*$.  Then consider the map
\begin{equation}
\label{1.latticemap}
\Phi_\cA : T_N \longrightarrow \C^s
\end{equation}
defined by
\[
\Phi_\cA(t) = \big(\chi^{m_1}(t),\dots,\chi^{m_s}(t)\big) \in \C^s.
\]

\begin{definition}
\label{1.latticeptdef}
Given a finite set $\cA \subseteq M$, the affine toric variety $\ya$ is
defined to be the Zariski closure of the image of the map $\Phi_\cA$
from \eqref{1.latticemap}.
\end{definition}

This definition is justified by the following proposition.

\begin{proposition}
\label{1.lattpt}
Given $\cA \subseteq M$ as above, let $\Z\cA \subseteq M$ be the
sublattice generated by $\cA$.  Then $\ya$ is an affine toric variety
whose torus has character lattice $\Z\cA$.  In particular, the dimension
of $\ya$ is the rank of $\Z\cA$.
\end{proposition}

\begin{proof}
The map \eqref{1.latticemap} can be regarded as a map
\[
\Phi_\cA : T_N \longrightarrow (\C^*)^s 
\]
of tori.  By Proposition~\ref{1.toriprops}, the image $T =
\Phi_\cA(T_N)$ is a torus that is closed in $(\C^*)^s$.  The
latter implies that $\ya \cap (\C^*)^s = T$ since $\ya$ is the Zariski
closure of the image.  It follows that the image is Zariski open in
$\ya$.  Furthermore, $T$ is irreducible (it is a torus), so the same
is true for its Zariski closure $\ya$.

We next consider the action of $T$.  Since $T \subseteq (\C^*)^s$, an
element $t \in T$ acts on $\C^s$ and takes varieties to varieties.
Then
\[
T = t\cdot T \subseteq t\cdot \ya
\]
shows that $t\cdot\ya$ is a variety containing $T$.  Hence $\ya
\subseteq t\cdot \ya$ by the definition of Zariski closure.  Replacing
$t$ with $t^{-1}$ leads to $\ya = t\cdot \ya$, so that the action of
$T$ induces an action on $\ya$.  We conclude that $\ya$ is an
affine toric variety.

It remains to compute the character lattice $M(T)$ of $T$.  Since $T =
\Phi_\cA(T_N)$, the map $\Phi_\cA$ gives the commutative diagram
\[
\SelectTips {cm}{}
\xymatrix{
T_N \ar[r]^(.44){\Phi_\cA} \ar@{->>}[rd] & (\C^*)^s \\
& T\ \ \  \ar@{^{(}->}[u]
}
\]
where $\twoheadrightarrow$ denotes a surjective map and
$\hookrightarrow$ an injective map.  This diagram of tori induces a
commutative diagram of character lattices
\[
\SelectTips {cm}{}
\xymatrix{
M  & \Z^s \ar[l]_(.45){\widehat{\Phi}_\cA} \ar@{->>}[d] \\
& M(T).\ \ \ \ar@{_{(}->}[lu]
}
\]
Since $\widehat{\Phi}_\cA:\Z^s \to M$ takes the standard basis
$e_1,\dots,e_s$ to $m_1,\dots,m_s$, the image of $\widehat{\Phi}_\cA$
is $\Z\cA$.  By the diagram, we obtain $M(T) \simeq \Z\cA$.  Then we
are done since the dimension of a torus equals the rank of its
character lattice.
\end{proof}

In concrete terms, fix a basis of $M$, so that we may assume $M =
\Z^n$.  Then the $s$ vectors in $\cA \subseteq \Z^n$ can be regarded as
the columns of an $n\times s$ matrix $A$ with integer entries.  In
this case, the the dimension of $\ya$ is simply the rank of the matrix
$A$.

We will see below that every affine toric variety is isomorphic to
$\ya$ for some finite subset $\cA$ of a lattice.

\subsection*{\emph{Toric Ideals}} Let $\ya \subseteq \C^s =
\Spec(\C[x_1,\dots,x_s])$ be the affine toric variety coming from a
finite set $\cA = \{m_1,\dots,m_s\} \subseteq M$.  We can describe the
ideal $\I(\ya) \subseteq \C[x_1,\dots,x_s]$ as follows.  As in the
proof of Proposition~\ref{1.lattpt}, \eqref{1.latticemap} induces a
map of character lattices
\[
\widehat{\Phi}_\cA: \Z^s \longrightarrow M
\]
that sends the standard basis $e_1,\dots,e_s$ to $m_1,\dots,m_s$.  Let
$L$ be the kernel of this map, so that we have an exact sequence
\[
0 \longrightarrow L \longrightarrow \Z^s \longrightarrow M.
\]
In down to earth terms, elements $\ell = (\ell_1,\dots,\ell_s)$ of $L$
satisfy $\sum_{i=1}^s \ell_i m_i = 0$ and hence record the linear
relations among the $m_i$.

Given $\ell = (\ell_1,\dots,\ell_s) \in L$, set
\[
\ell_+ = \sum_{\ell_i > 0} \ell_i e_i\quad \text{and}\quad
\ell_- = -\sum_{\ell_i < 0} \ell_i e_i.
\]
Note that $\ell = \ell_+ - \ell_-$ and that $\ell_+,\ell_- \in \N^s$.
It follows easily that the binomial
\[
x^{\ell_+} - x^{\ell_-} = {\textstyle \prod_{\ell_i > 0}
x_i^{\ell_i} - \prod_{\ell_i < 0} x_i^{-\ell_i}}
\]
vanishes on the image of \eqref{1.latticemap} and hence on
$\ya$ since $\ya$ is the Zariski closure of the image.  

\begin{proposition}
\label{1.toricid}
The ideal of the affine toric variety $\ya \subseteq \C^s$ is
\[
\I(\ya) = \big\langle x^{\ell_+} - x^{\ell_-} \mid \ell \in
L\big\rangle = \big\langle x^{\alpha} - x^{\beta} \mid
\alpha,\beta \in \N^s\ \text{\rm and } \alpha -\beta \in L\big\rangle.
\]
\end{proposition}

\begin{proof} We leave it to the reader to prove equality of the two
ideals on the right (Exercise~\ref{1.1.2}).  Let $I_L$ denote this
ideal and note that $I_L \subseteq \I(\ya)$.  We prove the opposite
inclusion following \cite[Lemma 4.1]{sturmfels}.  Pick a monomial
order $>$ on $\C[x_1,\dots,x_s]$ and an isomorphism $T_N \simeq
\torus$.  Thus we may assume $M = \Z^n$ and the map $\Phi : \torus \to
\C^s$ is given by Laurent monomials $t^{m_i}$ in variables
$t_1,\dots,t_n$.  If $I_L \ne \I(\ya)$, then we can pick $f \in
\I(\ya) \setminus I_L$ with minimal leading monomial $x^{\alpha} =
\prod_{i=1}^s x_i^{a_i}$.  Rescaling if necessary, $x^{\alpha}$
becomes the leading term of $f$.

Since $f(t^{m_1},\dots,t^{m_s})$ is identically zero as a polynomial
in $t_1,\dots,t_n$, there must be cancellation involving the term
coming from $x^{\alpha}$.  In other words, $f$ must contain a monomial
$x^{\beta} = \prod_{i=1}^s x_i^{b_i} < x^{\alpha}$ such that
\[
\prod_{i=1}^s (t^{m_i})^{a_i} = \prod_{i=1}^s (t^{m_i})^{b_i}. 
\]
This implies that
\[
\sum_{i=1}^s a_i m_i = \sum_{i=1}^s b_i m_i,
\]
so that $\alpha - \beta = \sum_{i=1}^s (a_i-b_i)e_i \in L$.  Then
$x^{\alpha} - x^{\beta} \in I_L$ by the second description of
$I_L$.  It follows that $f - x^{\alpha} + x^{\beta}$ also lies in
$\I(\ya) \setminus I_L$ and has strictly smaller leading term.  This
contradiction completes the proof.
\end{proof}

Given a finite set $\cA \subseteq M$, there are several methods to
compute the ideal $\I(\ya) = I_L$ of Proposition~\ref{1.toricid}.  For
simple examples, the rational implicitization algorithm of \cite[\S3
of Chapter 3]{iva} can be used.  It is also possible to compute $I_L$
using a basis of $L$ and ideal quotients (Exercise~\ref{1.1.3}).
Further comments on computing $I_L$ can be found in \cite[Chapter
12]{sturmfels}.

Inspired by Proposition~\ref{1.toricid}, we make the following
definition. 

\begin{definition}
\label{1.toriciddef}
Let $L \subseteq \Z^s$ be a sublattice.
\begin{thmenum}
\item The ideal $I_L = \big\langle x^{\alpha} - x^{\beta} \mid 
\alpha,\beta \in \N^s\ \text{\rm and } \alpha -\beta \in L\big\rangle$
is called a \emph{\bfseries lattice ideal}.
\item A prime lattice ideal is called a \emph{\bfseries toric ideal}.
\end{thmenum}
\end{definition}

Since toric varieties are irreducible, the ideals appearing in
Proposition~\ref{1.toricid} are toric ideals.  Examples of toric
ideals include:
\begin{align*}
\text{Example~\ref{1.cuspex}}:\ &\langle x^{3}-y^{2} \rangle
   \subseteq \C[x,y]\\ 
\text{Example~\ref{1.xyzw}}:\ &\langle xz-yw \rangle \subseteq
   \C[x,y,z,w]\\
\text{Example~\ref{1.rnccone}}:\ &\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].
\end{align*}
(The latter is the ideal of the rational normal cone $\widehat{C}_d
\subseteq \C^{d+1}$.)  In each example, we have a prime ideal
generated by binomials.  As we now show, such ideal are automatically
toric.

\begin{proposition}
\label{1.toricid2}
An ideal $I \subseteq \C[x_1,\dots,x_s]$ is toric if and only if it is
prime and generated by binomials.
\end{proposition}

\begin{proof}
One direction is obvious.  So suppose that $I$ is prime and generated
by binomials $x^{\alpha_i} - x^{\beta_i}$.  Then observe that
$\V(I)\cap(\C^*)^s$ is nonempty (it contains $(1,\dots,1)$) and is a
subgroup of $(\C^*)^s$ (easy to check).  Since $\V(I) \subseteq \C^s$
is irreducible, it follows that $\V(I)\cap(\C^*)^s$ is an irreducible
subvariety of $(\C^*)^s$ that is also a subgroup.  By
Proposition~\ref{1.toriprops}, we see that $T = \V(I)\cap(\C^*)^s$ is
a torus.

Projecting on the $i$th coordinate of $(\C^*)^s$ gives a character
$T \hookrightarrow (\C^*)^s \to \C^*$, which by our usual convention
we write as $\chi^{m_i} : T \to \C^*$ for $m_i \in M(T)$.  It follows
easily that $\V(I) = \ya$ for $\cA = \{m_1,\dots,m_s\}$, and since $I$
is prime, we have $I = \I(\ya)$ by the Nullstellensatz.  Then $I$ is
toric by Proposition~\ref{1.toricid}.
\end{proof}

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

\subsection*{A\emph{ffine Semigroups}} A \emph{semigroup} is a set
$\sS$ with an associative binary operation and an identity element.  To
be an \emph{affine semigroup}, we further require that
\begin{coxitem}
\item The binary operation on $\sS$ is commutative.  We will write the
  operation as $+$ and the identity element as $0$.  Thus a
  finite set $\cA \subseteq \sS$ gives
\[
\N\cA = \big\{{\textstyle\sum_{m\in \cA}} a_m m \mid a_m \in \N\big\}
\subseteq \sS.
\] 
\item The semigroup is finitely generated, meaning that there is a
  finite set $\cA \subseteq \sS$ such that $\N\cA = \sS$. 
\item The semigroup can be embedded in a lattice $M$.
\end{coxitem}
The simplest example of an affine semigroup is $\N^n \subseteq \Z^n$.
More generally, given a lattice $M$ and a finite set $\cA \subseteq M$,
we get the affine semigroup $\N\cA \subseteq M$.  Up to isomorphism,
all affine semigroups are of this form.

Given an affine semigroup $\sS \subseteq M$, the \emph{semigroup
algebra} $\C[\sS]$ is the vector space over $\C$ with $\sS$ as basis and
multiplication induced by the semigroup structure of $\sS$.  To make
this precise, we think of $M$ as the character lattice of a torus
$T_N$, so that $m \in M$ gives the character $\chi^m$.  Then
\[
\C[\sS] = \Big\{\sum_{m \in \sS} c_m \chi^m \mid c_m \in \C \text{ and }
c_m = 0 \text{ for all but finitely many } m\Big\},
\]
with multiplication induced by
\[
\chi^m\cdot \chi^{m'} = \chi^{m+m'}.
\]
If $\sS = \N\cA$ for $\cA = \{m_1,\dots,m_s\}$, then $\C[\sS] =
\C[\chi^{m_1},\dots,\chi^{m_s}]$.  

Here are two basic examples.

\begin{example}
\label{1.nnsg}
The affine semigroup $\N^n \subseteq \Z^n$ gives the polynomial ring
\[
\C[\N^n] = \C[x_1,\dots,x_n],
\]
where $x_i = \chi^{e_i}$ and $e_1,\dots,e_n$ is the standard basis of
$\Z^n$.\eoe
\end{example}

\begin{example}
\label{1.torussg}
If $e_1,\dots,e_n$ is a basis of a lattice $M$, then $M$ is generated
by $\cA = \{\pm e_1,\dots,\pm e_n\}$ as an affine semigroup.  Setting $t_i =
\chi^{e_i}$ gives the Laurent polynomial ring
\[
\C[M] = \C[t_1^{\pm1},\dots,t_n^{\pm1}].
\]
Using Example~\ref{1.torus}, one sees that $\C[M]$ is the
coordinate ring of the torus $T_N$.\eoe  
\end{example}

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

\begin{proposition}
\label{1.semigrp}
Let $\sS \subseteq M$ be an affine semigroup.  
\begin{thmenum}
\item $\C[\sS]$ is an integral domain and finitely generated as a
  $\C$-algebra.  
\item $\Spec(\C[\sS])$ is an affine toric variety whose torus has
  character lattice $\Z\sS$, and if $\sS = \N\cA$ for a finite set $\cA
  \subseteq M$, then $\Spec(\C[\sS]) = \ya$.
\end{thmenum}
\end{proposition}

\begin{proof}
As noted above, $\cA = \{m_1,\dots,m_s\}$ implies $\C[\sS] =
\C[\chi^{m_1},\dots,\chi^{m_s}]$, so $\C[\sS]$ is finitely generated.
Since $\C[\sS] \subseteq \C[M]$ follows from $\sS \subseteq M$, we see
that $\C[\sS]$ is an integral domain by Example~\ref{1.torussg}. 

Using $\cA = \{m_1,\dots,m_s\}$, we get the $\C$-algebra homomorphism 
\[
\pi : \C[x_1,\dots,x_s] \longrightarrow \C[M] 
\]
where $x_i \mapsto \chi^{m_i} \in M$. This corresponds to the morphism 
\[
\Phi_\cA : T_N \longrightarrow \C^s
\]
from \eqref{1.latticemap}, i.e., $\pi = (\Phi_\cA)^*$ in the notation
of \S0.  One checks that the kernel of $\pi$ is the toric ideal
$\I(\ya)$ (Exercise~\ref{1.1.4}).  Since the image of $\pi$ is
$\C[\chi^{m_1},\dots,\chi^{m_s}] = \C[\sS]$, the coordinate ring of
$\ya$ is
\begin{equation}
\label{1.yasg}
\begin{aligned}
\C[\ya] &= \C[x_1,\dots,x_n]/\I(\ya)\\ &=
\C[x_1,\dots,x_n]/\mathrm{Ker}(\pi) \simeq \mathrm{Im}(\pi) =
\C[\sS].
\end{aligned}
\end{equation}
Thus $\Spec(\C[\sS])$ is the toric variety $\ya$.  Since
$\sS = \N\cA$ implies $\Z\sS = \Z\cA$, the torus of $\ya =
\Spec(\C[\sS])$ has the desired character lattice by
Proposition~\ref{1.lattpt}. 
\end{proof}

Here is an example of this proposition.

\begin{example}
\label{1.cuspsg}
Consider the affine semigroup $\sS \subseteq \Z$ generated by $2$ and
$3$, so that $\sS = \{0,2,3,\dots\}$.  To study the semigroup algebra
$\C[\sS]$, we use \eqref{1.yasg}.  If $\cA = \{2,3\}$, then
$\Phi_\cA(t) = (t^2,t^3)$ and the toric ideal is $\I(\ya) = \langle
x^{3}-y^{2}\rangle$ by Example~\ref{1.cuspex}.  Hence
\[
\C[\sS] = \C[t^2,t^3] \simeq \C[x,y]/\langle x^{3}-y^{2}\rangle
\]
and the affine toric variety $\ya$ is the curve $x^{3}=y^{2}$.\eoe
\end{example}

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

\begin{theorem}
\label{1.affinethm}
Let $V$ be an affine variety.  The following are equivalent:
\begin{thmenum}
\item $V$ is an affine toric variety according to
  Definition~\ref{1.afftordef}.
\item $V = \ya$ for a finite set $\cA$ in a lattice.
\item $V$ is an affine variety defined by a toric ideal.
\item $V = \Spec(\C[\sS])$ for an affine semigroup $\sS$.
\end{thmenum}
\end{theorem}

\begin{proof}
The implications (b) $\Leftrightarrow$ (c) $\Leftrightarrow$ (d)
$\Rightarrow$ (a) follow from Propositions~\ref{1.lattpt},
\ref{1.toricid} and \ref{1.semigrp}.  For (a) $\Rightarrow$ (d), let
$V$ be an affine toric variety containing the torus $T_N$, and let $M$
be the character lattice of $T_N$.  Since the coordinate ring of $T_N$
is the semigroup algebra $\C[M]$, the inclusion $T_N \subseteq V$
induces a map of coordinate rings
\[
\C[V] \longrightarrow \C[M].
\]
This map is injective since $T_N$ is Zariski dense in $V$, so that we
can regard $\C[V]$ as a subalgebra of $\C[M]$.  Let
\[
\sS = \{m \in M \mid \chi^m \in \C[V]\}
\]
and note that $\sS$ is a semigroup.  We will show that $\sS$ is
finitely generated with semigroup algebra equal to $\C[V]$.  This will
complete the proof of the theorem.

The inclusion $\C[\sS] \subseteq \C[V]$ is obvious.  For the opposite
inclusion, pick $f \ne 0$ in $\C[V]$.  Using $\C[V] \subseteq \C[M]$, we
can write
\[
f = \sum_{m \in \cB} c_m \chi^m,
\]
where $\cB \subseteq M$ is finite and $c_m \ne 0$ for all $m \in \cB$.
Let 
\[
W = \Span(\chi^m \mid m \in \cB) \subseteq \mr
\]
be the subspace spanned by the $\chi^m$.  Thus $f \in W\cap \C[V]$.
If $t \in T_N$, then the action of $t$ on $V$ induces an action on the
coordinate ring $\C[V]$, so that we get $t\cdot f \in \C[V]$.  By
Definition~\ref{1.afftordef}, this extends the usual action of $t$ on
$T_N$, which in terms of the coordinate ring $\C[M]$ is given by
$t\cdot \chi^m = \chi^m(t) \chi^m$.  It follows that $W$ and hence
$W\cap \C[V]$ are stable under the action of $T_N$.  Since $W \cap
\C[V]$ is finite-dimensional, Proposition~\ref{1.torusaction} implies
that $W \cap \C[V]$ is spanned by simultaneous eigenvectors of~$T_N$.
But this is taking place in $\C[M]$, where the simultaneous
eigenvectors are the characters!  It follows that $W \cap \C[V]$ is
spanned by characters.  Then the above expression for $f \in W \cap
\C[V]$ implies that $\chi_m \in \C[V]$ for $m \in \cB$.  It follows
that $f \in \C[\sS]$, proving that $\C[V] = \C[\sS]$.

It remains to show that $\sS$ is finitely generated.  Since $\C[V]$ is
finitely generated, we can find $f_1,\dots,f_s \in \C[V]$ with $\C[V]
= \C[f_1,\dots,f_s]$.  Expressing each $f_i$ in terms of characters as
above gives the desired finite generating set of $\sS$.  Hence $\sS$
is an affine semigroup. 
\end{proof}

Here is one way to think about the above proof.  When an irreducible
affine variety $V$ contains a torus $T_N$ as a Zariski open subset, we
have the inclusion
\[
\C[V] \subseteq \C[M].
\]
Thus $\C[V]$ consists of those functions on the torus $T_N$ that
extend to polynomial functions on $V$.  Then the key insight is that
$V$ is a toric variety \emph{precisely when the functions that extend
are determined by the characters that extend}.

\begin{example}
\label{1.xyzw2}
We've seen that $V = \V(xy-zw) \subseteq \C^4$ is a toric variety
with toric ideal $\langle xy-zw\rangle \subseteq \C[x,y,z,w]$.
Also, the torus is $(\C^*)^3$ via the map $(t_1,t_2,t_3) \mapsto
(t_1,t_2,t_3, t_1t_2t_3^{-1})$.  The lattice points used in this map
can be represented as the columns of the matrix
\begin{equation}
\label{1.matxyzw}
\begin{pmatrix} 1 & 0 & 0 & \phantom{-}1\\ 0 & 1 & 0 & \phantom{-}1\\
  0 & 0 & 1 & -1
\end{pmatrix}.
\end{equation}
The corresponding semigroup $\sS \subseteq \Z^3$ consists of the
$\N$-linear combinations of the column vectors.  Hence the elements of
$\sS$ are lattice points lying in the polyhedral region in $\R^3$
pictured in Figure~1.  In this figure, the four
vectors generating $\sS$ are shown in bold, and the boundary of the
polyhedral region is
\begin{figure}[ht]
\label{1.fig1}
\begin{center}
\includegraphics[height=2.55in]{ch1fig1.ps}
\end{center}
\caption{Cone containing the lattice points corresponding to
  $V=\V(xy-zw)$} 
\end{figure}
partially shaded.  In the terminology of \S2, this polyhedral region
is a \emph{rational polyhedral cone}.  In Exercise~\ref{1.1.5} you
will show that $\sS$ consists of \emph{all} lattice points lying in
the cone in Figure~1.  We will use this in \S3 to prove that $V$ is
normal. \eoe
\end{example}

\exercises

\small

\begin{exercise}
\label{1.1.1}
As in Example~\ref{1.rnccone}, let
\[
I = \langle x_i x_{j+1} - x_{i+1} x_j \mid 0 \le i < j \le
   d-1\rangle \subseteq \C[x_0,\dots,x_d]
\]
and let $\widehat{C}_d$ be the surface parametrized by
\[
\Phi(s,t) = (s^d,s^{d-1}t,\dots,s t^{d-1},t^d) \in \C^{d+1}.
\]
\begin{thmenum}
\item Prove that $\V(I) = \Phi(\C^2) \subseteq \C^{d+1}$.  Thus
  $\widehat{C}_d = \V(I)$.
\item  Prove that $\I(\widehat{C}_d)$ is homogeneous.
\item Consider lex monomial order with $x_0 > x_1 > \cdots > x_d$.
  Let $f \in \I(\widehat{C}_d)$ be homogeneous of degree $\ell$ and
  let $r$ be the remainder of $f$ on division by the generators of
  $I$.  Prove that $r$ can be written
\[
r = h_0(x_0,x_1) + h_1(x_1,x_2) + \cdots + h_{d-1}(x_{d-1},x_d)
\]
where $h_i$ is homogeneous of degree $\ell$.  Also explain why we may
assume that the coefficient of $x_i^{\ell}$ in $h_i$ is zero for $1 \le
i \le d-1$.
\item Use part (c) and $r(s^d,s^{d-1}t,\dots,st^{d-1},t^d) = 0$ to
  show that $r = 0$.  
\item Use parts (b), (c) and (d) to prove that $I =
  \I(\widehat{C}_d)$.  Also explain why the generators of $I$ are a
  Gr\"obner basis for the above lex order.
\end{thmenum}  
\end{exercise}

\begin{exercise}
\label{1.1.2}
Let $L \subseteq \Z^s$ be a sublattice.  Prove that
\[
\langle x^{\ell_+} - x^{\ell_-} \mid \ell \in L\rangle = \langle
x^{\alpha} - x^{\beta} \mid \alpha, \beta \in \N^s,\ \alpha -
\beta \in L\rangle.
\]
Note that when $\ell \in L$, the vectors $\ell_+,\ell_- \in \N^s$ have
disjoint support (i.e., no coordinate is positive in both), while this
may fail for arbitrary $\alpha, \beta \in \N^s$ with $\alpha - \beta
\in L$.
\end{exercise}

\begin{exercise}
\label{1.1.3}
Let $I_L$ be a toric ideal and let $\ell^{1},\dots,\ell^{r}$ be
a basis of the sublattice $L \subseteq \Z^s$.  Define
\[
I_\mathbf{L} = \langle x^{\ell_+^{i}} - x^{\ell_-^{i}} \mid i
= 1,\dots,r\rangle.
\]
Prove that $I_L = I_\mathbf{L}\col \langle x_1\cdots
x_s\rangle^\infty$.  Hint:\ Given $\alpha, \beta \in \N^s$ with $\alpha
- \beta \in L$, write $\alpha - \beta = \sum_{i=1}^r a_i \ell^{i}$,
$a_i \in \Z$.  This implies
\[
x^{\alpha-\beta} - 1 = \prod_{a_i > 0}
\left(\frac{x^{\ell_+^{i}}}{x^{\ell_-^{i}}}\right)^{a_i}
\prod_{a_i < 0}
\left(\frac{x^{\ell_-^{i}}}{x^{\ell_+^{i}}}\right)^{-a_i} - 1.
\]
Show that multiplying this by $(x_1\cdots x_s)^k$ gives an element of
$I_\mathbf{L}$ for $k \gg 0$.  (By being more careful, one can show
that this result holds for lattice ideals.  See \cite[Lemma~7.6]{ms}.)
\end{exercise}

\begin{exercise}
\label{1.1.4}
Fix an affine variety $V$ and 
$f_1,\dots,f_s \in \C[V]$.  This gives a polynomial map $\Phi: V \to
\C^s$, which on coordinate rings is given by
\[
\Phi^* : \C[x_1,\dots,x_s] \longrightarrow \C[V],\qquad x_i \longmapsto
f_i. 
\]
Let $Y \subseteq \C^s$ be the Zariski closure of the image of $\Phi$.
\begin{thmenum}
\item Prove that $\I(Y) = \mathrm{Ker}(\Phi^*)$.
\item Explain how this applies to the proof of
  Proposition~\ref{1.semigrp}.
\end{thmenum} 
\end{exercise}

\begin{exercise}
\label{1.1.5}
Let $m_1 = (1,0,0), m_2 = (0,1,0), m_3 = (0,0,1), m_4 =
(1,1,-1)$ be the columns of the matrix in Example~\ref{1.xyzw2} and
let 
\[
C = \Big\{\sum_{i=1}^4 \lambda_i m_i \mid \lambda_i \in \R_{\ge
  0}\Big\} \subseteq \R^3
\]
be the cone in Figure~1.  Prove that $C\cap\Z^3$ is a
semigroup generated by $m_1,m_2,m_3,m_4$.
\end{exercise}

\begin{exercise}
\label{1.1.6}
An interesting observation is that different sets of lattice
points can parametrize the same affine toric variety, even though
these parametrizations behave slightly differently.  In this exercise
you will consider the parametrizations
\[
\Phi_1(s,t) = (s^2,st,st^3)\quad \text{and}\quad \Phi_2(s,t) =
(s^3,st,t^3).
\]
\begin{thmenum}
\item Prove that $\Phi_1$ and $\Phi_2$ both give the affine toric
variety $Y = \V(xz-y^3) \subseteq \C^3$.  
\item We can regard $\Phi_1$ and $\Phi_2$ as maps 
\[
\Phi_1 : \C^2 \longrightarrow Y \quad \text{and}\quad \Phi_2 : \C^2
\longrightarrow Y.
\]
Prove that $\Phi_2$ is surjective and that $\Phi_1$ is not.
\end{thmenum}
In general, a finite subset $\cA \subseteq \Z^n$ gives a rational map
$\Phi_\cA : \C^n \dashrightarrow \ya$.  The image of $\Phi_\cA$ in
$\C^s$ is called a \emph{toric set} in the literature.  Thus
$\Phi_1(\C^2)$ and $\Phi_2(\C^2)$ are toric sets.  
%When a toric set equals the corresponding affine toric variety is
%studied in \cite{kt} and \cite{rvz}.
The papers \cite{kt} and \cite{rvz} study when a toric set equals the
corresponding affine toric variety. 
\end{exercise}

\begin{exercise}
\label{1.1.7}
In Example~\ref{1.rnccone} and Exercise~\ref{1.1.1} we constructed the
rational normal cone $\widehat{C}_d$ using all monomials of degree $d$
in $s,t$.  If we drop some of the monomials, things become more
complicated.  For example, consider the surface parametrized by
\[
\Phi(s,t) = (s^4,s^3t,st^3,t^4) \in \C^4.
\]
This gives a toric variety $Y \subseteq \C^4$.  Show that the toric
ideal of $Y$ is given by
\[
\I(Y) = \langle xw-yz, yw^{2}-z^{3}, xz^{2}-y^{2}w,
x^{2}z-y^{3}\rangle \subseteq \C[x,y,z,w].
\]
The toric ideal for $\widehat{C}_4$ has quadratic generators; by
removing the monomial $s^2t^2$, we now get cubic generators.  In
Chapter~\ref{poly} we will use this example to construct a projective
curve that is normal but not projectively normal.
\end{exercise}

\begin{exercise}
\label{1.1.8}
Prove that a lattice ideal $I_L$ for $L \subseteq \Z^s$ is a
toric ideal if and only if $\Z^s/L$ is torsion-free.  Hint:\ When
$\Z^s/L$ is torsion-free, it can be regarded as the character lattice
of a torus.  The other direction of the proof is more challenging.  If
you get stuck, see \cite[Theorem~7.4]{ms}.
\end{exercise}

\begin{exercise}
\label{1.1.9}
Prove that $I = \langle x^{2}-1,xy-1,yz-1\rangle$ is the
lattice ideal for the lattice 
\[
L = \{(a,b,c) \in \Z^3 \mid a+b+c \equiv 0 \bmod 2\} \subseteq \Z^3.
\]
Also compute primary decomposition of $I$ to show that $I$ is not
prime.
\end{exercise}

\begin{exercise}
\label{1.1.10}
Let $T_N$ be a torus with character lattice $M$.  Then every
point $t \in T_N$ gives an evaluation map $\phi_t : M \to \C^*$
defined by $\phi_t(m) = \chi^m(t)$.  Prove that $\phi_t$ is a group
homomorphism and that the map $t \mapsto \phi_t$ induces a group
isomorphism
\[
T_N \simeq \mathrm{Hom}_\Z(M,\C^*).
\]
\end{exercise}

\begin{exercise}
\label{1.1.11}
Consider tori $T_1$ and $T_2$ with character lattices $M_1$ and $M_2$.
By Example~\ref{1.torussg}, the coordinate rings of $T_1$ and $T_2$
are $\C[M_1]$ and $\C[M_2]$.  Let $\Phi : T_1 \to T_2$ be a morphism
that is a group homomorphism.  Then $\Phi$ induces maps
\[
\widehat{\Phi} : M_2 \longrightarrow M_1\quad \text{and}\quad
\Phi^* : \C[M_2] \longrightarrow \C[M_1]
\]
by composition.  Prove that $\Phi^*$ is the map of semigroup algebras
induced by the map $\widehat{\Phi}$ of affine semigroups.
\end{exercise}

\begin{exercise}
\label{1.1.12}
A commutative semigroup $\sS$ is \emph{cancellative} if $u+v =
u+w$ implies $v = w$ for all $u,v,w \in \sS$ and \emph{torsion-free}
if $nu = 0$ implies $u = 0$ for all $n \in \N\setminus\{0\}$ and $u
\in \sS$.  Prove that $\sS$ is affine if and only if it is finitely
generated, cancellative, and torsion-free.
\end{exercise}

\begin{exercise}
\label{1.1.13}
The requirement that an affine semigroup be finitely generated is
important since lattices contain semigroups that are not finitely
generated.  For example, let $\tau > 0$ be irrational and consider the
semigroup
\[
\sS = \{(a,b) \in \N^2 \mid b \ge \tau a\} \subseteq \Z^2.
\]
Prove that $\sS$ is not finitely generated.  (When $\tau$ satisfies a
quadratic equation with integer coefficients, the generators of $\sS$
are related to continued fractions.  For example, when $\tau =
(1+\sqrt5)/2$ is the golden ratio, the minimal generators of $\sS$ are
$(1,0)$ and $(F_{2n-1},F_{2n})$ for $n = 1,2,\dots$, where $F_n$ is
the $n$th Fibonacci number.  See \cite{stark} for further details.)
\end{exercise}

\begin{exercise}
\label{1.1.14}
Suppose that $\phi : M \to M$ is a group isomorphism.  Fix
a finite set $\cA \subseteq M$ and let $\cB = \phi(\cA)$.  Prove that
the toric varieties $\ya$ and $Y_\cB$ are equivariantly isomorphic
(meaning that the isomorphism respects the torus action).
\end{exercise}

\normalsize

\section{Cones and Affine Toric Varieties} 

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

\subsection*{\emph{Polyhedral Cones}} Fix a pair of dual vector spaces
$\mr$ and $\nr$ of dimension $n$.  Our discussion will omit most
proofs---we refer the reader to \cite{fulton} for a more detailed
treatment and \cite[Appendix~A.1]{oda} for careful statements.  See
also \cite{bronsted,grunbaum,rockafellar}.

\begin{definition} 
\label{1.conedef}
A \emph{\bfseries polyhedral cone} in $\nr$ is a set of the form
\[
\sigma = \Cone(S)= \Big\{\sum_{u \in S} \lambda_u u
\mid \lambda_u \ge 0\Big\} \subseteq \nr,
\]
where $S \subseteq \nr$ is finite.  We say that $\sigma$ is
\emph{\bfseries generated} by $S$.  Also set $\Cone(\emptyset) =
\{0\}$.
\end{definition}

One easily checks that polyhedral cone $\sigma$ is \emph{convex},
meaning that $x,y \in \sigma \Rightarrow \lambda x + (1-\lambda) y \in
\sigma$ for all $0 \le \lambda \le 1$, and is a \emph{cone}, meaning
$x \in \sigma \Rightarrow \lambda x \in \sigma$ for all $\lambda \ge
0$.  Cones satisfying Definition~\ref{1.conedef} are sometimes called
``convex polyhedral cones'', but we will drop the ``convex'' since all
cones we consider will be convex.

Examples of convex polyhedral cones include the first quadrant in
$\R^2$ or first octant in $\R^3$.  For another example, the cone
$\Cone(e_1,e_2,e_1+e_3.e_2+e_3) \subseteq \R^3$  is
pictured in Figure~2.  
\begin{figure}[ht]
\begin{center}
\includegraphics[height=2.5in]{ch1fig2.ps}
\end{center}
\caption{Cone in $\R^3$ generated by $e_1,e_2,e_1+e_3.e_2+e_3$}
\end{figure}
It is also possible to have cones that contain entire lines.
For example, $\Cone(e_1,-e_1) \subseteq \R^2$ is the $x$-axis, while
$\Cone(e_1,-e_1,e_2)$ is the closed upper half-plane $\{(x,y) \in \R^2
\mid y \ge 0\}$.  As we will see below, these last two examples are
not \emph{strongly convex}.

We can also create cones using \emph{polytopes}, which are defined as
follows. 

\begin{definition}
\label{1.polydef}
A \emph{\bfseries polytope} in $\nr$ is a set of the
form
\[
P = \Conv(S) = \Big\{ \sum_{u \in S} \lambda_u u
\mid \lambda_u \ge 0,\ \sum_{u\in S} \lambda_u = 1\Big\} \subseteq \nr,
\]
where $S \subseteq \nr$ is finite.  We say that $P$ is the
\emph{\bfseries convex hull} of $S$.
\end{definition}

Polytopes include all polygons in $\R^2$ and polyhedra in $\R^3$.  As
we will see in later chapters, polytopes play a prominent role in the
theory of toric varieties.  Here, however, we simply observe that a
polytope $P \subseteq \nr$ gives a polyhedral cone in $\nr\times\R$ by
taking the cone
\[
\sigma = \{\lambda\cdot(u,1) \in \nr\times\R \mid u \in P,\ \lambda
\ge 0\}. 
\]
If $P = \Conv(S)$, then we can also describe this as $\sigma =
\Cone(S\times\{1\})$.  Figure~3 on the next page shows what this looks
when $P$ is a pentagon in the plane
\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.8in]{ch1fig3.ps}
\end{center}
\caption{Cone over a pentagon $P \subseteq \R^2$}
\end{figure}

The \emph{dimension} of a polyhedral cone $\sigma$, denoted
$\dim\sigma$, is the dimension of the smallest subspace $W$ of $\nr$
containing $\sigma$.  We call $W$ the \emph{span} of $\sigma$.

\subsection*{\emph{Dual Cones and Faces}} As usual, the pairing
between $\mr$ and $\nr$ is denoted $\langle \ ,\ \rangle$.  

\begin{definition}
\label{1.dualconedef}
Given a polyhedral cone $\sigma \subseteq \nr$, its \emph{\bfseries dual
cone} is defined by
\[
\sigma^\vee = \{m \in \mr \mid \langle m,u\rangle \ge 0 \text{ for 
all } u \in \sigma\}.
\]
\end{definition}

Duality has the following important properties.

\begin{proposition}
\label{1.dualprops}
Let $\sigma \subseteq \nr$ be a polyhedral cone.  Then $\sigma^\vee$ is
a polyhedral cone in $\mr$ and $(\sigma^\vee)^\vee = \sigma$.\qed
\end{proposition}

Given $m \ne 0$ in $\nr$, we get the hyperplane
\[
H_m = \{ u \in \mr \mid \langle m,u\rangle = 0\}
\]
and the closed half-space 
\[
H_m^+ = \{ u \in \mr \mid \langle m,u\rangle \ge 0\}.
\]
We say that $H_m$ is a \emph{supporting hyperplane} of a polyhedral
cone $\sigma$ if $\sigma \subseteq H_m^+$, and $H_m^+$ is a
\emph{supporting half-space}.  Note that $H_m$ is a supporting
hyperplane of $\sigma$ if and only if $m \in \sigma^\vee \setminus
\{0\}$.  Furthermore, if $m_1,\dots,m_s$ generate $\sigma^\vee$, then
it is straightforward to check that 
\begin{equation}
\label{1.conehalfspace}
\sigma = H^+_{m_1} \cap \cdots \cap H^+_{m_1}.
\end{equation}
Thus every polyhedral cone is an intersection of finitely many closed
half-spaces. 

We can use supporting hyperplanes and half-spaces to define
\emph{faces} of a cone.

\begin{definition}
\label{1.facedef}
A \emph{\bfseries face} of the polyhedral cone $\sigma$ is $\tau =
H_m\cap\sigma$ for some $m \in \sigma^\vee$.  Using $m = 0$ shows that
$\sigma$ as a face of itself.  Faces $\tau \ne \sigma$ are called
\emph{\bfseries proper faces}.
\end{definition}

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

\begin{lemma}
\label{1.faceprops}
Let $\sigma = {\rm Cone}(S)$ be a polyhedral cone.  Then:
\begin{thmenum}
\item Every face of $\sigma$ is a polyhedral cone.
\item An intersection of two faces of $\sigma$ is again a
face of $\sigma$.
\item A face of a face of $\sigma$ is again a face of $\sigma$.\qed
\end{thmenum}
\end{lemma}

A \emph{facet} of $\sigma$ is a face $\tau$ of codimension 1, i.e.,
$\dim\tau = \dim\sigma - 1$.  An \emph{edge} of $\sigma$ is a face of
dimension 1.  In Figure~4 we illustrate a 3-dimensional cone with
shaded facets and a supporting hyperplane (a plane in this case) that
cuts out the vertical edge of the cone.
\begin{figure}[ht]
\begin{center}
\includegraphics[height=2.2in]{ch1fig4.ps}
\end{center}
\caption{A cone $\sigma \subseteq \R^3$ with shaded facets and a
  supporting hyperplane} 
\end{figure}

Here are some properties of facets.

\begin{proposition}
\label{1.facetprops}
Let $\sigma \subseteq \nr \simeq \R^n$ be a polyhedral cone.  Then:
\begin{thmenum}
\item If $\dim\sigma = n$ and the facets of $\sigma$ are $\tau_i =
  H_{m_i} \cap \sigma$ for $m_i \in \sigma^\vee$, $1 \le i \le s$,
  then
\[
  \sigma = H^+_{m_1} \cap \cdots \cap H^+_{m_s}\quad \text{and}\quad
  \sigma^\vee = \Cone(m_1,\dots,m_s).
\]
\item Every proper face $\tau$ of $\sigma$ is the intersection of the
facets of $\sigma$ containing $\tau$.\qed
\end{thmenum}
\end{proposition}

Note how part (a) of the proposition refines \eqref{1.conehalfspace}
when $\dim \sigma = \dim\nr$.  

When working in $\R^n$, dot product allows us to identify the dual
with $\R^n$.  {}From this point of view, the vectors $m_1,\dots,m_s$ in
part (a) of the proposition are \emph{facet normals}, i.e.,
perpendicular to the facets.  This makes it easy to compute examples.

\begin{example}
\label{1.dualconeex}
It easy to see that the facet normals to the cone $\sigma
\subseteq \R^3$ in Figure~2 are $m_1 = e_1, m_2 = e_2, m_3 = e_3, m_4 =
e_1+e_2-e_3$.  Hence
\[
\sigma^\vee = \Cone(e_1, e_2, e_3, e_1+e_2-e_3) \subseteq \R^3.
\]
This is the cone of Figure~1 at the end of \S1 whose lattice points
describe the semigroup of the affine toric variety $\V(xy-zw)$
(see Example~\ref{1.xyzw2}).  As we will see, this is part of how
cones describe normal affine toric varieties.

Now consider $\sigma^\vee$, which is the cone in Figure~1.  The reader
can check that the facet normals of this cone are $e_1, e_2, e_1+e_3,
e_2+e_3$.  Using duality and part (b) of
Proposition~\ref{1.facetprops}, we obtain
\[
\sigma = (\sigma^\vee)^\vee = \Cone(e_1,e_2,e_1+e_3,e_2+e_3).
\]
Hence we recover our original description of $\sigma$.\eoe
\end{example}

In this example, facets of the cone correspond to edges of its dual.
More generally, given a face $\tau$ of a polyhedral cone $\sigma
\subseteq \nr$, we define
\begin{align*}
\tau^\perp &= \{m \in \mr \mid \langle m,u\rangle = 0 \text{ for all }
u \in \tau\}\\ 
\tau^* &= \{m \in \sigma^\vee \mid \langle m,u\rangle = 0 \text{ for
all } u \in \tau\}\\
&= \sigma^\vee \cap \tau^\perp.
\end{align*}
We call $\tau^*$ the \emph{dual face} of $\tau$ because of the
following proposition.

\begin{proposition}
\label{1.dualfaceprops}
If $\tau$ is a face of a polyhedral cone $\sigma$ and $\tau^* =
\sigma^\vee\cap\tau^\perp$, then:
\begin{thmenum}
\item $\tau^*$ is a face of $\sigma^\vee$.  
\item The map $\tau \mapsto \tau^*$ is a bijective
inclusion-reversing correspondance between the faces of $\sigma$ and
the faces of $\sigma^\vee$.
\item $\dim\tau + \dim\tau^* = n$.\qed
\end{thmenum}
\end{proposition}

Here is an example of Proposition~\ref{1.dualfaceprops} when
$\dim\sigma < \dim\nr$.

\begin{example}
\label{1.notmaxdim}
Let $\sigma = \Cone(e_1,e_2) \subseteq \R^3$.  Figure~5 
shows $\sigma$ and $\sigma^\vee$.  
\begin{figure}[ht]
\begin{center}
\includegraphics[height=2.3in]{ch1fig5.ps}
\end{center}
\caption{A $2$-dimensional cone $\sigma \subseteq \R^3$ and its dual
  $\sigma^\vee \subseteq \R^3$}  
\end{figure}
You should check that the maximal face of $\sigma$, namely $\sigma$
itself, gives the minimal face $\sigma^*$ of $\sigma^\vee$, namely the
$z$-axis.  Note also that
\[
\dim \sigma + \dim\sigma^* = 3
\]
even though $\sigma$ has dimension $2$. \eoe
\end{example}

\subsection*{\emph{Strong Convexity}} Of the cones shown in Figures
1--5, all but $\sigma^\vee$ in Figure~5 have the nice property that the
origin is a face.  Such cones are called \emph{strongly convex}.  This
condition can be stated several ways.

\begin{proposition}
\label{1.stcx}
Let $\sigma \subseteq \nr \simeq \R^n$ be a polyhedral cone.  Then:
\begin{align*}
\hspace*{18pt}\sigma \text{ is strongly convex} &\iff \{0\} \text{ is
  a face of } 
  \sigma\\ 
&\iff \sigma \text{ contains no positive-dimensional subspace}\\
&\iff \sigma\cap(-\sigma) = \{0\}\\
&\iff \dim\sigma^\vee = n. \hspace*{146truept}\qed
\end{align*}  
\end{proposition}

You will prove Proposition~\ref{1.stcx} in Exercise~\ref{1.2.1}.  One
corollary is that if a polyhedral cone $\sigma$ is strongly convex of
maximal dimension, then so is $\sigma^\vee$.  The cones pictured in
Figures~1--4 satisfy this condition.

In general, a polyhedral cone $\sigma$ always has a minimal face that
is the largest subspace $W$ contained in $\sigma$.  Furthermore, we
have $W = \sigma \cap (-\sigma) = (\sigma^\vee)^\perp$, and
$\overline{\sigma} = \sigma/W \subseteq \nr/W$ is a strongly convex
polyhedral cone.  See Exercise~\ref{1.2.2}.

\subsection*{\emph{Rational Polyhedral Cones}} Let $N$ and
$M$ be dual lattices with associated vector spaces $\nr = N\otimes_\Z
\R$ and $\mr = M\otimes_\Z \R$.  For $\R^n$ we usually use the lattice
$\Z^n$.

\begin{definition}
\label{1.ratpolycone}
A polyhedral cone $\sigma \subseteq \nr$ is \emph{\bfseries rational} if
$\sigma = \Cone(S)$ for some finite set $S \subseteq N$.
\end{definition}

The cones appearing in Figures 1, 2 and 5 are rational.  We note
without proof that faces and duals of rational polyhedral cones are
rational.  Furthermore, if $\sigma = \Cone(S)$ for $S \subseteq N$
finite and $N_\Q = N\otimes_\Z \Q$, then
\begin{equation}
\label{1.coneoverQ}
\sigma\cap N_\Q = \big\{{\textstyle\sum_{u \in S}} \lambda_u u
\mid \lambda_u \ge 0 \text{ in } \Q\big\}.
\end{equation}

One new feature is that a strongly convex rational polyedral cone
$\sigma$ has a canonical generating set, constructed as follows.  Let
$\rho$ be a edge of $\sigma$.  Since $\sigma$ is strongly convex,
$\rho$ is a \emph{ray}, i.e., a half-line, and since $\rho$ is
rational, the semigroup $\rho\cap N$ is generated by a unique element
$u$ of the intersection.  We call $u$ the \emph{ray generator} of
$\rho$.  Figure~6 on the next page shows the ray generator of a
rational ray $\rho$ in the plane.  The dots are the lattice $N = \Z^2$
and the white ones are $\rho\cap N$.
\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.4in]{ch1fig6.ps}
\end{center}
\caption{A rational ray $\rho \subseteq \R^2$ and its unique ray generator}  
\end{figure}

\begin{lemma}
\label{1.raygens}
A strongly convex rational polyedral cone is generated by the ray
generators of its edges.\qed 
\end{lemma}

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

In a similar way, a rational polyhedral cone $\sigma$ of maximal
dimension has unique \emph{facet normals}, which are the ray
generators of the dual $\sigma^\vee$, which is strongly convex by
Proposition~\ref{1.stcx}.

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

\begin{definition} 
\label{1.regsimp}
Let $\sigma \subseteq \nr$ be a strongly convex rational polyhedral
cone.
\begin{thmenum}
\item $\sigma$ is \emph{\bfseries smooth} or \emph{\bfseries regular}
if its minimal generators form part of a $\Z$-basis of $N$,
\item $\sigma$ is \emph{\bfseries simplicial} if its minimal
generators are linearly independent over $\R$.
\end{thmenum}
\end{definition}

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

\subsection*{\emph{Semigroup Algebras and Affine Toric Varieties}}
Given a rational polyhedral cone $\sigma \subseteq \nr$, the lattice
points
\[
\sS_\sigma = \sigma^\vee \cap M \subseteq M
\]
form a semigroup.  A key fact is that this semigroup is finitely
generated.

\begin{proposition}[Gordan's Lemma]
\label{1.gordan}
$\sS_\sigma = \sigma^\vee \cap M$ is finitely generated and hence is
an affine semigroup.
\end{proposition}

\begin{proof}
Since $\sigma^\vee$ is rational polyhedral, 
$\sigma^\vee = {\rm Cone}(T)$ for a finite set $T \subseteq M$.  Then $K
= \{\sum_{m\in T} \delta_m m \mid 0 \le \delta_u < 1\}$ is a
bounded region of $\mr \simeq \R^n$, so that $K\cap M$ is finite since
$M \simeq \Z^n$.  Note that $T \cup (K\cap M) \subseteq
\sS_\sigma$.

We claim $T \cup (K\cap M)$ generates $\sS_\sigma$ as a semigroup.  To
prove this, take $w \in \sS_\sigma$ and write $w = \sum_{m\in T}
\lambda_m m$ where $\lambda_m \ge 0$.  Then $\lambda_m = \lfloor
\lambda_m\rfloor + \delta_m$ with $\lfloor \lambda_m\rfloor \in \N$
and $0 \le \delta_m < 1$, so that
\[
w = \sum_{m\in T} \lfloor \lambda_m\rfloor m + \sum_{m\in T}
\delta_m m.
\]
The second sum is in $K\cap M$ (remember $w \in M$).  It follows that
$w$ is a nonnegative integer combination of elements of $T \cup (K\cap
M)$.
\end{proof}

{}From \S1, we know that affine semigroups give affine toric varieties.
Hence we get the following result.

\begin{theorem}
\label{1.coneafftor}
Let $\sigma \subseteq \nr \simeq \R^n$ be a rational polyhedral cone
with semigroup $\sS_\sigma = \sigma^\vee\cap M$.  Then
\[
U_\sigma = \Spec(\C[\sS_\sigma]) = \Spec(\C[\sigma^\vee\cap M])
\]
is an affine toric variety.  Furthermore,
\[
\dim U_\sigma = n \iff \text{the torus of } U_\sigma \text{ is } T_N
\iff \sigma \text{ is strongly convex}.
\]
\end{theorem}

\begin{proof}
By Gordan's Lemma and Proposition~\ref{1.semigrp}, $U_\sigma$ is an
affine toric variety whose torus has character lattice $\Z\sS_\sigma
\subseteq M$.  To study $\Z\sS_\sigma$, note that
\[
\Z\sS_\sigma = \sS_\sigma - \sS_\sigma = \{m_1 - m_2 \mid m_1,m_2 \in
\sS_\sigma\}.
\]
Now suppose that $k m \in \Z\sS_\sigma$ for some $k > 1$ and $m \in
M$.  Then $k m = m_1 - m_2$ for $m_1,m_2 \in \sS_\sigma =
\sigma^\vee\cap M$.  Since $m_1$ and $m_2$ lie in the convex set
$\sigma^\vee$, we have
\[
m + m_2 = {\textstyle\frac1k}m_1 + {\textstyle\frac{k-1}k}m_2 \in
\sigma^\vee.
\]
It follows that $m = (m+m_2)-m_2 \in \Z\sS_\sigma$, so that
$M/\Z\sS_\sigma$ is torsion-free.  Hence
\begin{equation}
\label{1.zss}
\text{the torus of } U_\sigma \text{ is } T_N \iff \Z\sS_\sigma = M
\iff \mathrm{rank}\,\Z\sS_\sigma = n.
\end{equation}
Since $\sigma$ is strongly convex if and only if $\dim\sigma^\vee = n$
(Proposition~\ref{1.stcx}), it remains to show that
\[
\dim U_\sigma = n \iff \mathrm{rank}\,\Z\sS_\sigma = n \iff
\dim\sigma^\vee = n.
\]
The first equivalence follows since the dimension of an affine toric
variety is the dimension of its torus, which is the rank of its
character lattice.  We leave the proof of the second equivalence to
the reader (Exercise~\ref{1.2.3}).
\end{proof}

Since we want our affine toric varieties to contain the torus $T_N$,
we consider only those affine toric varieties $U_\sigma$ for which
$\sigma \subseteq \nr$ is strongly convex.

Our first example of Theorem~\ref{1.coneafftor} is an affine toric
variety we know well.

\begin{example}
\label{1.xyzw3}
Let $\sigma = \Cone(e_1,e_2,e_1+e_3,e_2+e_3) \subseteq \nr = \R^3$ with
$N = \Z^3$.  This is the cone pictured in Figure~2.  By
Example~\ref{1.dualconeex}, $\sigma^\vee$ is the cone pictured in
Figure~1, and by Example~\ref{1.xyzw2}, the lattice points in this
cone are generated by columns of matrix \eqref{1.matxyzw}.  It follows
from Example~\ref{1.xyzw2} that $U_\sigma$ is the affine toric variety
$\V(xy-zw)$. \eoe
\end{example}

Here are two further examples of Theorem~\ref{1.coneafftor}.

\begin{example}
\label{1.smoothcone}
Fix $0 \le r \le n$ and set $\sigma = \Cone(e_1,\dots,e_r) \subseteq
\R^n$.  Then
\[
\sigma^\vee = \Cone(e_1,\dots,e_r,\pm e_{r+1},\dots,\pm e_n)
\]
and the corresponding affine toric variety is
\[
U_\sigma =
\Spec(\C[x_1,\dots,x_r,x_{r+1}^{\pm1},\dots,x_n^{\pm1}]) = \C^r
\times (\C^*)^{n-r}
\]
(Exercise~\ref{1.2.4}).  This implies the general fact that if $\sigma
\subseteq \nr \simeq \R^n$ is a smooth cone of dimension $r$, then
$U_\sigma \simeq \C^r \times (\C^*)^{n-r}$.  \eoe
\end{example}

Figure~5 illustrates the cones involved in Example~\ref{1.smoothcone}
when $r = 2$ and $n = 3$.

\begin{example}
\label{1.rnccone2}
Fix a positive integer $d$ and let $\sigma = \Cone(e_1,de_1-e_2)
\subseteq \R^2$.  This has dual cone $\sigma^\vee =
\Cone(e_1,e_1+de_2)$.  Figure~7 shows $\sigma^\vee$
when $d = 4$.  The affine semigroup $S_\sigma = \sigma^\vee \cap
\Z^2$ is generated by the lattice points $(1,i)$ for $0 \le i \le d$.
When $d=4$, these are the white dots in Figure~7.  (You will prove
these assertions in Exercise~\ref{1.2.5}.)
\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.95in]{ch1fig7.ps}
\caption{The cone $\sigma^\vee$ when $d = 4$}  
\end{center}
\end{figure}

By \S1, the affine toric variety $U_\sigma$ is the Zariski closure of
the image of the map $\Phi : (\C^*)^2 \to \C^{d+1}$ defined by
\[
\Phi(s,t) = (s,st,st^2,\dots,st^d).
\]
This map has the same image as the map $(s,t) \mapsto (s^d,s^{d-1}t,
\dots,st^{d-1},t^d)$ used in Example~\ref{1.rnccone}.  Thus
$U_\sigma$ is isomorphic to the rational normal cone $\widehat{C}_d
\subseteq \C^{d+1}$ whose ideal is generated by the $2\time2$ minors
of the matrix
\[
\begin{pmatrix} x_0 & x_1 & \cdots & x_{d-2} & x_{d-1} \\
x_1 & x_2 & \cdots & x_{d-1} & x_{d}
\end{pmatrix}.
\]
Note that the cones $\sigma$ and $\sigma^\vee$ are simplicial but not
smooth.  \eoe
\end{example}

We will return to this example often as we develop the theory of toric
varieties.  One thing evident in Example~\ref{1.rnccone} is the
difference between \emph{cone generators} and \emph{semigroup
generators}:\ the cone $\sigma^\vee$ has two generators but the
semigroup $S_\sigma = \sigma^\vee \cap \Z^2$ has $d+1$. 

When $\sigma \subseteq \nr$ has maximal dimension, the semigroup
$S_\sigma = \sigma^\vee \cap M$ has a unique minimal generating set
constructed as follows.  Define an element $m \ne 0$ of $\sS_\sigma$
to be \emph{irreducible} if $m = m'+m''$ for $m',m'' \in \sS_\sigma$
implies $m' = 0$ or $m'' = 0$.

\begin{proposition}
\label{1.hilbbasis}
Let $\sigma \subseteq \nr$ be strongly convex of maximal dimension and
let $\sS_\sigma = \sigma^\vee \cap M$.  Then
\[
\cH = \{m \in \sS_\sigma \mid m \text{ is irreducible}\}
\]
has the following properties:
\begin{thmenum}
\item $\cH$ is finite and generates $\sS_\sigma$.
\item $\cH$ contains the ray generators of the edges of $\sigma^\vee$.
\item $\cH$ is the minimal generating set of $\sS_\sigma$ with respect
  to inclusion.
\end{thmenum}
\end{proposition}

\begin{proof} 
Proposition~\ref{1.stcx} implies that $\sigma^\vee$ is strongly
convex, so we can find an element $u \in \sigma\cap N\setminus\{0\}$
such that $\langle m,u \rangle \in \N$ for all $m \in \sS_\sigma$ and
$\langle m,u \rangle = 0$ if and only if $m = 0$.

Now suppose that $m \in \sS$ is not irreducible.  Then $m = m' + m''$
where $m'$ and $m''$ are nonzero elements of $\sS_\sigma$.  It follows
that
\[
\langle m,u \rangle = \langle m',u \rangle + \langle m'',u \rangle
\]
with $\langle m',u \rangle, \langle m'',u \rangle \in
\N\setminus\{0\}$, so that 
\[
\langle m',u \rangle < \langle m,u \rangle \quad\text{and}\quad
\langle m'',u \rangle < \langle m,u \rangle.
\]
Using induction on $\langle m,u\rangle$, we conclude that every
element of $\sS_\sigma$ is a sum of irreducible elements, so that
$\cH$ generates $\sS_\sigma$.  Furthermore, using a finite generating
set of $\sS_\sigma$, one easily sees that $\cH$ is finite.  This
proves part (a).  The remaining parts of the proof are covered in
Exercise~\ref{1.2.6}. 
\end{proof} 

The set $\cH \subseteq \sS_\sigma$ is called the \emph{Hilbert basis} of
$\sS_\sigma$ and its elements are the \emph{minimal generators} of
$\sS_\sigma$.  Algorithms for computing Hilbert bases are discussed in
\cite[Section 7.3]{ms} and \cite[Chapter 13]{sturmfels}.

\exercises

\small

\begin{exercise}
\label{1.2.1}
Prove Proposition~\ref{1.stcx}.
\end{exercise}

\begin{exercise}
\label{1.2.2}
Let $\sigma \subseteq \nr$ be a polyhedral cone.
\begin{thmenum}
\item Use Proposition~\ref{1.dualfaceprops} to prove that $\sigma$
  has a unique minimal face with respect to inclusion.  Let $W$ denote
  this minimal face.
\item Prove that $W = (\sigma^\vee)^\perp$.
\item Prove that $W$ is the largest subspace contained in $\sigma$.
\item Prove that $W = \sigma\cap(-\sigma)$.
\item Prove that $\overline{\sigma} = \sigma/W \subseteq \nr/W$ is a
strongly convex polyhedral cone.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{1.2.3}
Fix a lattice $M \simeq \R^n$ and let $\Span(S)$ denote the
span over $\R$ of a subset $S \subseteq \mr$.
\begin{thmenum}
\item Let $S \subseteq M$ be finite.  Prove that $\mathrm{rank}\,\Z S
  = \dim \Span(S)$.
\item Let $S \subseteq \mr$ be finite.  Prove that $\dim \Cone(S) = \dim
  \Span(S)$. 
\item Use parts (a) and (b) to complete the proof of
  Theorem~\ref{1.coneafftor}. 
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{1.2.4}
Prove the assertions made in Example~\ref{1.smoothcone}.
\end{exercise}

\begin{exercise}
\label{1.2.5}
Prove the assertions made in Example~\ref{1.rnccone2}.  Hint:\ First
show that when a cone is smooth, the ray generators of the cone also
generate the corresponding semigroup.  Then write the cone
$\sigma^\vee$ of Example~\ref{1.rnccone2} as a union of such cones.
\end{exercise}

\begin{exercise}
\label{1.2.6}
Complete the proof of Proposition~\ref{1.hilbbasis}.  Hint for
part (b):\ Show that the ray generators of the edges of $\sigma^\vee$
are irreducible in $\sS_\sigma$.  Given an edge $\rho$ of
$\sigma^\vee$, it will help to pick $u \in \sigma\cap N\setminus\{0\}$
such that $\rho = H_u\cap \sigma^\vee$.
\end{exercise}

\begin{exercise}
\label{1.2.7}
Let $\sigma \subseteq \nr$ be a cone generated by a set of
linearly independent vectors in $\nr$.  Show that $\sigma$ is strongly
convex and simplicial.  
\end{exercise}

\begin{exercise}
\label{1.2.8}
Explain the picture illustrated in Figure 8 in terms of
Proposition~\ref{1.facetprops}. 
{\normalsize\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.07in]{ch1fig8.ps}
\end{center}
\caption{A cone $\sigma$ in the plane and its dual}  
\end{figure}}
\end{exercise}

\begin{exercise}
\label{1.2.9}
Let $P \subseteq \nr$ be a polytope lying in an affine hyperplane
(= translate of a hyperplane) not containing the origin.  Generalize
Figure~3 by showing that $P$ gives a convex polyhedral cone in $\nr$.
Draw a picture.
\end{exercise}

\begin{exercise}
\label{1.2.10}
Consider the cone $\sigma = \Cone(3e_1-2e_2,e_1) \subseteq \R^2$.
\begin{thmenum}
\item Describe $\sigma^\vee$ and find generators of $\sigma^\vee
\cap \Z^2$.  Draw a picture similar to Figure 7.
\item Compute the toric ideal of the affine toric variety $U_\sigma$
  and explain how this exercise relates to Exercise~\ref{1.1.6}.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{1.2.11}
Consider the simplicial cone $\sigma =
\Cone(e_1,e_2,e_1+e_2+2e_3) \subseteq \R^3$.   
\begin{thmenum}
\item Describe $\sigma^\vee$ and find generators of $\sigma^\vee \cap
\Z^3$.
\item Compute the toric ideal of the affine toric variety $U_\sigma$.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{1.2.12}
Let $\sigma$ be a strongly convex polyhedral cone of maximal
dimension.  Here is an example taken from \cite[p.~132]{fulton} to
show that $\sigma$ and $\sigma^\vee$ need not have the same number of
edges.  Let $\sigma \subseteq \R^4$ be the cone generated by $2e_i +
e_j$ for all $1 \le i, j \le 4$, $i \ne j$.
\begin{thmenum}
\item Show that $\sigma$ has 12 edges.
\item Show that $\sigma^\vee$ is generated by $e_i$ and $-e_i +
  2\sum_{j \ne i} e_j$, $1 \le i \le 4$ and has 8 edges.
\end{thmenum}
\end{exercise}

\normalsize

\section{Properties of Affine Toric Varieties}
 
The final task of this chapter is to explore the properties of
affine toric varieties.  We will also study maps between affine toric
varieties. 

\subsection*{\emph{Points of Affine Toric Varieties}} We first
consider various ways to describe the points of an affine toric
variety.

\begin{proposition}
\label{1.points}
Let $V = \Spec(\C[\sS])$ be the affine toric variety of the affine
semigroup $\sS$.  Then there are bijective correspondences between the
following objects: 
\begin{thmenum}
\item Points $p \in V$.
\item Maximal ideals $\mf \subseteq \C[\sS]$.
\item Semigroup homomorphisms $\sS \to \C$, where $\C$ is considered
  as a semigroup under multiplication.
\end{thmenum}
\end{proposition}

\begin{proof}
The correspondence between (a) and (b) is standard (see \cite[Theorem
5 of Chapter 5]{iva}).  The correspondence between (a) and (c) is
special to the toric case.

Given a point $p \in V$, define $\sS \to \C$ by sending $m \in \sS$ to
$\chi^m(p) \in \C$.  This makes sense since $\chi^m \in \C[\sS] =
\C[V]$.  One easily checks that $\sS \to \C$ is a semigroup
homomorphism.

Going the other way, let $\gamma : \sS \to \C$ be a semigroup
homomorphism.  Since $\{\chi^m\}_{m\in \sS}$ is a basis of $\C[\sS]$,
$\gamma$ induces a surjective linear map $\C[\sS] \to \C$ which is a
$\C$-algebra homomorphism.  The kernel is a maximal ideal and thus
gives a point $p \in V$ by the correspondence between (a) and (b).

We construct $p$ concretely as follows.  Let $\cA = \{m_1,\dots,m_s\}$
generate $\sS$, so that $V = \ya \subseteq \C^s$.  Let $p =
(\gamma(m_1),\dots,\gamma(m_s)) \in \C^s$.  Let us prove that $p \in
V$.  By Proposition~\ref{1.toricid}, it suffices to show that
$x^{\alpha} - x^{\beta}$ vanishes at $p$ for all exponent vectors
$\alpha = (a_1,\dots,a_s)$ and $\beta = (a_1,\dots,a_s)$ satisfying
\[
\sum_{i=1}^s a_i m_i = \sum_{i=1}^s b_i m_i.
\]
This is easy, since $\gamma$ being a semigroup homomorphism
implies that
\[
\prod_{i=1}^s \gamma(m_i)^{a_i} = \gamma\Big(\sum_{i=1}^s
a_i m_i\Big) = \gamma\Big(\sum_{i=1}^s b_i m_i\Big) = \prod_{i=1}^s
\gamma(m_i)^{b_i}.
\]
It is straightforward to show that this point of $V$ agrees with the
one constructed in the previous paragraph (Exercise~\ref{1.3.1}).
\end{proof}

As an application of this result, we describe the torus action on $V$.
In terms of the embedding $V = \ya \subseteq \C^s$, the proof of
Proposition~\ref{1.lattpt} shows that the action of $T_N$ on $\ya$ is
induced by the usual action of $(\Cstar)^s$ on $\C^s$.  But how do we
see the action instrinsically, without embedding into affine space?
This is where semigroup homomorphisms prove their value.  Fix $t \in
T_N$ and $p \in V$, and let $p$ correspond to the semigroup homorphism
$m \mapsto \gamma(m)$.  In Exercise~\ref{1.3.1} you will show that
$t\cdot p$ is given by the semigroup homomorphism $m \mapsto \chi^m(t)
\gamma(m)$.  This abstract description will prove useful in
Chapter~\ref{normal} when we study the orbits of the torus action.

From the point of view of group actions, the action of $T_N$ on $V$ is
given by a map $T_N \times V \to V$.  Since both sides are affine
varieties, this should be a morphism, meaning that it should come from
a $\C$-algebra homomorphism 
\[
\C[\sS] = \C[V] \longrightarrow \C[T_N \times V] = \C[T_N] \otimes_\C
\C[V] = \C[M] \otimes_\C \C[\sS].
\]
This homomorphism is given by $\chi^m \mapsto \chi^m \otimes \chi^m$
for $m \in \sS$ (Exercise~\ref{1.3.torusaction}).

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

\begin{proposition}
\label{1.fixedpoint}
Let $V$ be the affine toric variety.  Then:
\begin{thmenum}
\item If we write $V = \Spec(\C[\sS])$, then the torus action has a
fixed point if and only if $\sS$ is pointed, in which case the
unique fixed point is given by the the semigroup homomorphism $\sS \to
\C$ defined by
\begin{equation}
\label{1.fixptsg}
m \longmapsto 
\begin{cases}
1 & m = 0\\ 0 & m \ne 0.
\end{cases}
\end{equation}
\item If we write $V = \ya \subseteq \C^s$ for $\cA \subseteq \sS
\setminus \{0\}$, then the torus action has a fixed point if and only
$0 \in \ya$, in which case the unique fixed point is $0$.
\end{thmenum}
\end{proposition}

\begin{proof}
For part (a), let $p \in V$ be represented by the semigroup
homomorphism $\gamma : \sS \to \C$.  Then $p$ is fixed by the torus
action if and only if $\chi^m(t) \gamma(m) = \gamma(m)$ for all $m
\in \sS$ and $t \in T_N$.  This equation is satisfied for $m = 0$
since $\gamma(0) = 1$, and if $m \ne 0$, then picking $t$ with
$\chi^m(t) \ne 0$ shows that $\gamma(m) = 0$.  Thus, if a fixed point
exists, then it is unique and is given by \eqref{1.fixptsg}.  Then we
are done since \eqref{1.fixptsg} is a semigroup homomorphism if and
only if $\sS$ is pointed.

For part (b), first assume that $V = \ya \subseteq \C^s$ has a fixed
point, in which case $\sS = \N\cA$ is pointed and the unique point $p$
is given by \eqref{1.fixptsg}.  Then $\cA \subseteq \sS \setminus \{0\}$
and the proof of Proposition~\ref{1.points} imply that $p$ is the
origin in $\C^s$, so that $0 \in \ya$.  The converse follows since $0
\in \C^s$ is fixed by $(\Cstar)^s$ hence  by $T_N \subseteq (\Cstar)^s$.
\end{proof} 

When $V = U_\sigma$, we can state Proposition~\ref{1.fixedpoint} as
follows (Exercise~\ref{1.3.1a}).

\begin{corollary} 
\label{1.fixptcone}
Let $\sigma \subseteq \nr$ be a strongly convex polyhedral cone.  Then
the torus action on $U_\sigma$ has a fixed point if and only if
$\dim\sigma = \dim\nr$, in which case the fixed point is unique and is
given by the maximal ideal
\[
\langle \chi^m \mid m \in \sS_\sigma\setminus\{0\} \rangle \subseteq
\C[\sS_\sigma], 
\]
where as usual $\sS_\sigma = \sigma^\vee \cap M$.\qed
\end{corollary}

We will see in Chapter~\ref{normal} that this corollary is part of the
correspondence between torus orbits of $U_\sigma$ and faces of $\sigma$.

\subsection*{\emph{Normality and Saturation}} We next study the
question of when an affine toric variety $V$ is normal.  We need one
definition before stating our normality criterion.

\begin{definition}
\label{1.satdef}
An affine semigroup $\sS \subseteq M$ is \emph{\bfseries saturated} if
for all $k \in \N\setminus\{0\}$ and $m \in M$, $k m \in \sS$ implies
$m \in \sS$.
\end{definition}

For example, if $\sigma \subseteq \nr$ is a strongly convex rational
polyhedral cone, then $\sS_\sigma = \sigma^\vee \cap M$ is easily seen
to be saturated (Exercise~\ref{1.3.2}).  

\begin{theorem}
\label{1.satthm}
Let $V$ be an affine toric variety with torus $T_N$.  Then the
following are equivalent:
\begin{thmenum}
\item $V$ is normal.
\item $V = \Spec(\C[\sS])$, where $\sS \subseteq M$ is a saturated affine
semigroup.
\item $V = \Spec(\C[S_\sigma])\ (= U_\sigma)$, where $\sS_\sigma =
\sigma^\vee \cap M$ and $\sigma \subseteq \nr$ is a strongly convex
rational polyhedral cone.
\end{thmenum}
\end{theorem}

\begin{proof} By Theorem~\ref{1.affinethm}, $V = \Spec(\C[\sS])$ for
an affine semigroup $\sS$ contained in a lattice, and by
Proposition~\ref{1.semigrp}, the torus of $V$ has the character
lattice $M = \Z\sS$.  Also let $n = \dim V$, so that $M \simeq
\Z^n$.  We will use this to prove (a) $\Rightarrow$ (b)
$\Rightarrow$ (c). 

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

(b) $\Rightarrow$ (c):\ Let $\cA \subseteq \sS$ be a finite generating
set of $\sS$.  Then $\sS$ lies in the rational polyhedral cone
$\Cone(\cA) \subseteq \mr$, and $\mathrm{rank}\,\Z\cA = n$ implies
$\dim\Cone(\cA) = n$ by Exercise~\ref{1.2.3}.  It follows that $\sigma
= \Cone(\cA)^\vee \subseteq \nr$ is a strongly convex rational
polyhedral cone such that $\sS \subseteq \sigma^\vee \cap M$.  In
Exercise~\ref{1.3.2} you will prove that equality holds when $\sS$ is
saturated.  Hence $\sS = \sS_\sigma$.

(c) $\Rightarrow$ (a):\ We need to show that $\C[\sS_\sigma] =
\C[\sigma^\vee\cap M]$ is normal when $\sigma \subseteq \nr$ is a
strongly convex rational polyhedral cone.  Let $\rho_1,\dots,\rho_r$
be the rays of $\sigma$.  Since $\sigma$ is generated by its rays
(Lemma~\ref{1.raygens}), we have
\[
\sigma^\vee = \bigcap_{i=1}^r \rho_i^\vee,
\]
and intersecting with $M$ gives
\[
\sS_\sigma = \bigcap_{i=1}^r \sS_{\rho_i}.
\]
By Exercise~\ref{1.0.5}, $S_\sigma$ is normal if each $\sS_{\rho_i}$ is
normal, so it suffices to prove that $\sS_{\rho}$ is normal when
$\rho$ is a rational ray in $\nr$.  Let $u \in \rho\cap N$ be the ray
generator of $\rho$.  Since $u$ is \emph{primitive}, i.e, $\frac1k u
\notin N$ for all $k > 1$, we can find a basis $e_1,\dots,e_n$ of $N$
with $u = e_1$ (Exercise~\ref{1.3.3}).  Thus we can assume that $\rho
= \Cone(e_1)$, so that
\[
\C[\sS_\rho] = \C[x_1,x_2^{\pm1},\dots,x_n^{\pm1}]
\]
by Example~\ref{1.smoothcone}. But $\C[x_1,\dots,x_n]$ is normal (it
is a UFD), so its localization
\[
\C[x_1,\dots,x_n]_{x_2\cdots x_n} =
\C[x_1,x_2^{\pm1},\dots,x_n^{\pm1}] 
\]
is also normal by Exercise~\ref{1.0.5}.  This completes the proof.
\end{proof}

\begin{example}
\label{1.xyzw4}
We saw in Example~\ref{1.xyzw3} that $V = \V(xy-zw)$ is the affine
toric variety $U_\sigma$ of the cone $\sigma =
\Cone(e_1,e_2,e_1+e_3,e_2+e_3)$ pictured in Figure~1.  Then
Theorem~\ref{1.satthm} implies that $V$ is normal, as claimed in
Example~\ref{1.xyzw}.\eoe
\end{example}

\begin{example}
\label{1.rncnormal}
By Example~\ref{1.rnccone2}, the rational normal cone $\widehat{C}_d
\subseteq \C^{d+1}$ is the affine toric variety of $\sigma =
\Cone(de_1-e_2,e_1) \subseteq \R^2$ and hence is normal by
Theorem~\ref{1.satthm}.

It is instructive to view this example using the parametrization
\[
\Phi_\cA(s,t) = (s^d,s^{d-1}t,\dots,st^{d-1},t^d)\
\]
from Example~\ref{1.rnccone}.  Plotting the lattice points in $\cA$
for $d = 2$ gives the white squares in Figure~9 (a).
These generate the semigroup $\sS = \N\cA$, and the proof of
Theorem~\ref{1.satthm} gives the cone $\sigma^\vee = \Cone(e_1,e_2)$,
which is the first quadrant in the figure.
\begin{figure}[h]
\begin{center}
\includegraphics[height=1.8in]{ch1fig9.ps}
\end{center}
\caption{Lattice points for the rational normal cone $\widehat{C}_2$}  
\end{figure}

At first glance, something seems wrong.  The affine variety
$\widehat{C}_2$ is normal, yet in Figure~9 (a) the semigroup generated
by the white squares misses some lattice points in $\sigma^\vee$.
This semigroup does not look saturated.  How can the affine toric
variety be normal?

The problem is that we are using the wrong lattice!
Proposition~\ref{1.lattpt} tells us to use the lattice $\Z\cA$, which
gives the white dots and squares in Figure~9 (b).  This figure shows
that the white squares generate the semigroup of lattice points in
$\sigma^\vee$.  Hence $\sS$ is saturated and everything is fine.  \eoe
\end{example}

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

\subsection*{\emph{The Normalization of an Affine Toric Variety}}
The normalization of an affine toric variety is easy to describe.  Let
$V = \Spec(\C[\sS])$ for an affine semigroup~$\sS$, so that the torus
of the $V$ has character lattice $M = \Z\sS$.  Let $\Cone(\sS)$ denote
the cone of any finite generating set of $\sS$ and set $\sigma =
\Cone(\sS)^\vee \subseteq \nr$.  In Exercise~\ref{1.3.4} you will
prove the following.

\begin{proposition}
\label{1.normalize}
The above cone $\sigma$ is a strongly convex rational polyhedral cone
in $\nr$ and the inclusion $\C[\sS] \subseteq \C[\sigma^\vee\cap M]$
induces a morphism $U_\sigma \to V$ that is the normalization map of
$V$.\qed
\end{proposition}

The normalization of an affine toric variety of the form $\ya$ is
constructed by applying Proposition~\ref{1.normalize} to the affine
semigroup $\N\cA$ and the lattice $\Z\cA$.

\begin{example}
\label{1.nonnormalquartic}
Let $\cA = \{(4,0),(3,1),(1,3),(0,4)\} \subseteq \Z^2$.  Then
\[
\Phi_\cA(s,t) = (s^4,s^3t,st^3,t^4)
\]
parametrizes the surface $\ya \subseteq \C^4$ considered in
Exercise~\ref{1.1.7}.  This is almost the rational normal cone
$\widehat{C}_4$, except that we have omitted $s^2t^2$.
Using $(2,2) = \frac12\big((4,0)+(0,4)\big)$, we see that $\N\cA$ is
not saturated, so that $\ya$ is not normal.

Applying Proposition~\ref{1.normalize}, one sees that the
normalization of $\ya$ is $\widehat{C}_4$.  This is an affine variety
in $\C^5$, and the normalization map is induced by the obvious
projection $\C^5 \to \C^4$.\eoe
\end{example}

In Chapter~\ref{normal} we will show that the normalization map
$U_\sigma \to V$ constructed in Proposition~\ref{1.normalize} is
one-to-one and onto.

\subsection*{\emph{Smooth Affine Toric Varieties}} Our next goal is to
characterize when an affine toric variety is smooth.  Since smooth
affine varieties are normal (Proposition~\ref{1.smoothnor}), we need
only consider toric varieties $U_\sigma$ coming from strongly convex
rational polyhedral cones $\sigma \subseteq \nr$.

We begin by studying $U_\sigma$ when $\sigma$ has maximal dimension.
Then $\sigma^\vee$ is strongly convex, so that $\sS_\sigma =
\sigma^\vee \cap M$ has a Hilbert basis $\cH$.  Furthermore,
Corollary~\ref{1.fixptcone} tells us that torus action on $U_\sigma$
has a unique fixed point, denoted here by $p_\sigma \in U_\sigma$.
The point $p_\sigma$ and the Hilbert basis $\cH$ are related as
follows.

\begin{lemma} 
\label{1.maxideal}
Let $\sigma \subseteq \nr$ be a strongly convex rational polyhedral
cone of maximal dimension and let $T_{p_\sigma}(U_\sigma)$ be the
Zariski tangent space to the affine toric variety $U_\sigma$ at the
above point $p_\sigma$.  Then $\dim T_{p_\sigma}(U_\sigma) = |\cH|$.
\end{lemma}

\begin{proof}
By Corollary~\ref{1.fixptcone}, the maximal ideal of $\C[S_\sigma]$
corresponding to $p_\sigma$ is $\mf = \langle \chi^m \mid m \in
\sS_\sigma \setminus \{0\}\rangle$.  Since $\{\chi^m\}_{m \in
  \sS_\sigma}$ is basis of $\C[S_\sigma]$, we obtain
\[
\mf = \bigoplus_{m \ne 0}\ \C\chi^m 
= \!\bigoplus_{m \text{ irreducible}} \!\!\!\C\chi^m \ \ \oplus
    \!\!\bigoplus_{m \text{ reducible}}\!\! \C\chi^m
= \Big(\bigoplus_{m \in \cH} \C\chi^m \Big) \oplus \mf^2.
\]
It follows that $\dim \mf/\mf^2 = |\cH|$.  To relate this
to the maximal ideal $\mf_{U_\sigma,p_\sigma}$ in the local ring
$\cO_{U_\sigma,p_\sigma}$, we use the natural map
\[
\mf/\mf^2 \longrightarrow
\mf_{U_\sigma,p_\sigma}/\mf_{U_\sigma,p_\sigma}^2
\]
which is always an isomorphism (Exercise~\ref{1.3.5}).  Since
$T_{p_\sigma}(U_\sigma)$ is the dual space of
$\mf_{U_\sigma,p_\sigma}/\mf_{U_\sigma,p_\sigma}^2$, we see that $\dim
T_{p_\sigma}(U_\sigma) = |\cH|$.  
\end{proof}

The Hilbert basis $\cH$ of $S_\sigma$ gives $U_\sigma = Y_{\cH}
\subseteq \C^s$, where $s = |\cH|$.  This affine embedding is
especially nice.  Given \emph{any} affine embedding $U_\sigma
\hookrightarrow \C^\ell$, we have $\dim T_{p_\sigma}(U_\sigma) \le
\ell$ by Lemma~\ref{1.computeZar}.  In other words, $\dim
T_{p_\sigma}(U_\sigma)$ is a lower bound on the dimension of an affine
embedding.  Then Lemma~\ref{1.maxideal} shows that when $\sigma$ has
maximal dimension, the Hilbert basis of $\sS_\sigma$ gives the most
efficient affine embedding of $U_\sigma$.

\begin{example}
\label{1.rncemb}
In Example~\ref{1.rnccone2}, we saw that the rational normal cone
$\widehat{C}_d \subseteq \C^{d+1}$ is the toric variety coming from
$\sigma = \Cone(e_1,de_1-e_2) \subseteq \R^2$ and that $\sS_\sigma =
\sigma^\vee\cap\Z^2$ is generated by $(1,i)$ for $0 \le i \le d$.
These generators form the Hilbert basis of $\sS_\sigma$, so that the
Zariski tangent space of $0 \in \widehat{C}_d$ has dimension $d+1$.
Hence $\C^{d+1}$ in the smallest affine space in which we can embed
$\widehat{C}_d$.\eoe
\end{example}

We now come to our main result about smoothness.  Recall from \S2 that
a rational polyhedral cone is \emph{smooth} if it can be generated by
a subset of a basis of the lattice.

\begin{theorem}
\label{1.smoothaff}
Let $\sigma \subseteq \nr$ be a strongly convex polyhedral cone.  Then
$U_\sigma$ is smooth if and only if $\sigma$ is smooth.  Furthermore, 
all smooth affine toric varieties are of this form.  
\end{theorem}

\begin{proof}
If an affine toric variety is smooth, then it is normal by
Proposition~\ref{1.smoothnor} and hence of the form $U_\sigma$.  Also,
Example~\ref{1.smoothcone} implies that if $\sigma$ is smooth as a
cone, then $U_\sigma$ is smooth as a variety.  It remains to prove the
converse.  So fix $\sigma \subseteq \nr$ such that $U_\sigma$ is
smooth.  Let $n = \dim U_\sigma = \dim \nr$.

First suppose that $\sigma$ has dimension $n$ and let $p_\sigma \in
U_\sigma$ be the point studied in Lemma~\ref{1.maxideal}.  Since
$p_\sigma$ is smooth in $U_\sigma$, the Zariski tangent space
$T_{p_\sigma}(U_\sigma)$ has dimension $n$ by
Definition~\ref{1.smoothdef}.  On the other hand,
Lemma~\ref{1.maxideal} implies that $\dim T_{p_\sigma}(U_\sigma)$ is
the cardinality of the Hilbert basis $\cH$ of $\sS_\sigma =
\sigma^\vee \cap M$.  Thus
\[
n = |\cH| \ge |\{\text{edges } \rho \subseteq \sigma^\vee\}| \ge n,
\]
where the first inequality holds by Proposition~\ref{1.hilbbasis}
(each edge $\rho \subseteq \sigma^\vee$ contributes an element of
$\cH$) and the second holds since $\dim \sigma^\vee = n$.  It follows
that $\sigma$ has~$n$ edges and $\cH$ consists of the ray generators
of these edges.  Since $M = \Z\sS_\sigma$ by~\eqref{1.zss}, the $n$
edge generators of $\sigma^\vee$ generate the lattice $M \simeq \Z^n$
and hence form a basis of $M$.  Thus $\sigma^\vee$ is smooth, and then
$\sigma = (\sigma^\vee)^\vee$ is smooth since duality preserves
smoothness.

Next suppose that $\dim \sigma = r < n$.  We reduce to the previous
case as follows.  Let $N_1 \subseteq N$ be the smallest saturated
sublattice containing the generators of $\sigma$.  Then $N/N_1$ is
torsion-free, which by Exercise~\ref{1.3.3} implies the existence of a
sublattice $N_2 \subseteq N$ with $N = N_1 \oplus N_2$.  Note that
$\mathrm{rank}\,N_1 = r$ and $\mathrm{rank}\,N_2 = n-r$.

The cone $\sigma$ lies in both $(N_1)_\R$ and $\nr$.  This gives
affine toric varieties $U_{\sigma,N_1}$ and $U_{\sigma,N}$ of
dimensions $r$ and $n$ respectively.  Furthermore, $N = N_1 \oplus
N_2$ induces $M = M_1 \oplus M_2$, so that $\sigma \subseteq (N_1)_\R$
and $\sigma \subseteq \nr$ give the affine semigroups $\sS_{\sigma,N_1}
\subseteq M_1$ and $\sS_{\sigma,N} \subseteq M$ respectively.  It is
straighforward to show that
\[
\sS_{\sigma,N} = \sS_{\sigma,N_1} \oplus M_2,
\]
which in terms of semigroup algebras can be written
\[
\C[\sS_{\sigma,N}] \simeq \C[\sS_{\sigma,N_1}] \otimes_\C \C[M_2].
\]
The right-hand side is the coordinate ring of $U_{\sigma,N_1}\! \times
T_{N_2}$.  Thus
\begin{equation}
\label{1.proddecomp}
U_{\sigma,N} \simeq U_{\sigma,N_1}\! \times T_{N_2},
\end{equation}
which in turn implies that
\[
U_{\sigma,N} \simeq U_{\sigma,N_1}\! \times (\C^*)^{n-r} \subseteq
U_{\sigma,N_1}\! \times \C^{n-r}.
\]
Since we are assuming that $U_{\sigma,N}$ is smooth, it follows that
$U_{\sigma,N_1}\! \times \C^{n-r}$ is smooth at any point $(p,q)$ in
$U_{\sigma,N_1}\! \times (\C^*)^{n-r}$.  In Exercise~\ref{1.3.sm1} you
will show that
\begin{equation}
\label{1.secondshow}
U_{\sigma,N_1}\! \times \C^{n-r} \text{ is smooth at } (p,q)\
\Longrightarrow \ U_{\sigma,N_1} \text{ is smooth at } p.
\end{equation}
Letting $p = p_{\sigma} \in U_{\sigma,N_1}$, the previous case implies
that $\sigma$ is smooth in $N_1$ since $\dim \sigma = \dim (N_1)_\R$.
Hence $\sigma$ is clearly smooth in $N = N_1\oplus N_2$.
\end{proof}

\subsection*{\emph{Equivariant Maps between Affine Toric Varieties}}
We next study maps $V_1 \to V_2$ between affine toric varieties that
respect the torus actions on $V_1$ and $V_2$.  

\begin{definition}
\label{1.toricmapdef}
Let $V_i = \Spec(\C[\sS_i])$ be the affine toric varieties coming from
the affine semigroups $\sS_i$, $i=1,2$.  Then a morphism $\varphi :
V_1 \to V_2$ is \emph{\bfseries toric} if the corresponding map of
coordinate rings $\varphi^* : \C[\sS_2] \to \C[\sS_1]$ is induced by a
semigroup homomorphism $\widehat{\varphi} : \sS_2 \to \sS_1$.
\end{definition}

Here is our first result concerning toric morphisms.

\begin{proposition}
\label{1.toricmapprop}
Let $T_{N_i}$ be the torus of the affine toric variety $V_i$, $i=1,2$.
Then:
\begin{thmenum}
\item A morphism $\varphi : V_1 \to V_2$ is toric if and only if
\[
\varphi(T_{N_1}) \subseteq T_{N_2}
\]
and $\varphi\res{T_{N_1}} : T_{N_1} \to T_{N_2}$ is a group
homomorphism.   
\item A toric morphism $\varphi : V_1 \to V_2$ is {\bfseries
equivariant}, meaning that 
\[
\varphi(t\cdot p) = \varphi(t)\cdot \varphi(p)
\]
for all $t \in T_{N_1}$ and $p \in V_1$.
\end{thmenum}
\end{proposition}

\begin{proof}
Let $V_i = \Spec(\C[\sS_i])$, so that the character lattice of
$T_{N_i}$ is $M_i = \Z\sS_i$.  If $\varphi$ comes from a semigroup
homomorphism $\widehat{\varphi} : \sS_2 \to \sS_1$, then
$\widehat{\varphi}$ extends to a group homomorphism $\widehat{\varphi}
: M_2 \to M_1$ and hence gives a commutative diagram
\[
\begin{matrix}
\C[\sS_2] & \stackrel{\varphi^*}{\longrightarrow} &
\C[\sS_1]\\
\downarrow && \downarrow\\
\C[M_2] & \longrightarrow &
\C[M_1].
\end{matrix}
\]
Applying $\Spec$, we see that $\varphi(T_{N_1}) \subseteq T_{N_2}$,
and $\varphi\res{T_{N_1}} : T_{N_1} \to T_{N_2}$ is a group
homomorphism since $T_{N_i} = \mathrm{Hom}_\Z(M_i,\Cstar)$ by
Exercise~\ref{1.1.10}.  Conversely, if $\varphi$ satisfies these
conditions, then $\varphi\res{T_{N_1}} : T_{N_1} \to T_{N_2}$ induces
a diagram as above where the bottom map comes from a group
homomorphism $\widehat{\varphi} : M_2 \to M_1$.  This, combined with
$\varphi^*(\C[\sS_2]) \subseteq \sS_1$, implies that
$\widehat{\varphi}$ induces a semigroup homomorphism
$\widehat{\varphi} : \sS_2 \to \sS_1$.  This proves part (a) of the
proposition.

For part (b), suppose that we have a toric map $\varphi: V_1 \to V_2$.
The action of $T_{N_i}$ on $V_i$ is given by a morphism $\Phi_i :
T_{N_i}\! \times V_i \longrightarrow V_i$, and equivariance means that
we have a commutative diagram
\[
\SelectTips {cm}{}
\xymatrix{
T_{N_1}\! \times V_1 \ar[r]^(.6){\Phi_1}
 \ar[d]_{\varphi\smallres{T_{N_1}}\!\times
  \varphi} &  V_1 \ar[d]^{\varphi}\\
T_{N_2}\! \times V_2 \ar[r]^(.6){\Phi_2}  &  V_2.
}
\]
If we replace $V_i$ with $T_{N_i}$ in the diagram, then it certainly
commutes since $\varphi\res{T_{N_1}}$ is a group homomorphism.  Then
the whole diagram commutes since $T_{N_1}\! \times T_{N_1}$ is Zariski
dense in $T_{N_1}\! \times V_1$.
\end{proof}

We can also characterize toric morphisms between affine toric
varieties coming from strongly convex rational polyhedral cones.
First note that a homomorphism $\bar{\varphi} : N_1 \to N_2$ of
lattices gives a group homomorphism $\varphi : T_{N_1} \to T_{N_2}$ of
tori.  This follows from $T_{N_i} = N_i \otimes_\Z \Cstar$, and one
sees that $\varphi$ is a morphism.  Also, tensoring $\bar{\varphi}$
with $\R$ gives $\bar{\varphi}_\R : (N_1)_\R \to (N_2)_\R$

Here is the result, whose proof we leave to the reader
(Exercise~\ref{1.3.conemorphismexer}).

\begin{proposition} 
\label{1.3.conemorphism}
Suppose that we have strongly convex rational polyhedral cones $\sigma_i
\subseteq (N_i)_\R$ and a homomorphism $\bar{\varphi} : N_1 \to
N_2$.  Then $\varphi: T_{N_1} \to T_{N_2}$ extends to a map of affine
toric varieties $\varphi: U_{\sigma_1} \to U_{\sigma_2}$ if and only
if $\bar{\varphi}_\R(\sigma_1) \subseteq \sigma_2$.\qed
\end{proposition}

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

\subsection*{\emph{Faces and Affine Open Subsets}} Let $\sigma \subset
\nr$ be a strongly convex rational polyhedral cone and let $\tau
\subseteq \sigma$ be a face.  Then we can find $m \in \sigma^\vee \cap
M$ such that $\tau = H_m \cap \sigma$.  This allows us to relate
semigroup algebras of $\sigma$ and $\tau$ as follows.

\begin{proposition}
\label{1.facesg}
Let $\tau = H_m \cap \sigma$, $m \in \sigma^\vee \cap M$, be a face of
$\sigma$ as above.  Then the semigroup algebra $\C[\sS_\tau] =
\C[\tau^\vee \cap M]$ is the localization of $\C[\sS_\sigma] =
\C[\sigma^\vee \cap M]$ at $\chi^m \in \C[\sS_\sigma]$.  In other
words,
\[
\C[\sS_\tau] = \C[\sS_\sigma]_{\chi^m}.
\]
\end{proposition}

\begin{proof}
The inclusion $\tau \subseteq \sigma$ implies $\sS_\sigma \subset
\sS_\tau$, and since $\langle m,u \rangle = 0$ for all $u \in \tau$,
we have $\pm m \in \tau^\vee$.  It follows that
\[
\sS_\sigma + \Z(-m) \subseteq \sS_\tau.
\]
This inclusion is actually an equality, as we now prove.  Fix a finite
set $S \subset M$ with $\sigma = \Cone(S)$ and pick $m' \in
\sS_\tau$.  Set 
\[
C = \max_{u \in S}\{|\langle m',u\rangle|\} \in \N.
\]
It is straightforward to show that $m' + Cm \in \sS_\sigma$.  This
proves that
\[
\sS_\sigma + \Z(-m) = \sS_\tau,
\]
from which $\C[\sS_\tau] = \C[\sS_\sigma]_{\chi^m}$ follows
immediately. 
\end{proof}

We can interpret this in terms of toric morphisms as follows.  by
Proposition~\ref{1.3.conemorphism}, the identity map $N \to N$ and the
inclusion $\tau \subseteq \sigma$ give the toric morphism $U_\tau \to
U_\sigma$ that corresponds to the inclusion $\C[\sS_\sigma]) \subseteq
\C[\sS_\tau]$.  By Proposition~\ref{1.facesg},
\begin{equation}
\label{1.faceopen}
U_\tau \hskip-.4pt = \hskip-.4pt \Spec(\C[\sS_\tau])\hskip-.4pt =
\hskip-.4pt \Spec(\C[\sS_\sigma]_{\chi^m})\hskip-.4pt = \hskip-.4pt
\Spec(\C[\sS_\sigma])_{\chi^m}\hskip-.4pt = \hskip-.4pt
(U_\sigma)_{\chi^m} \hskip-.4pt \subseteq \hskip-.4pt U_\sigma.
\end{equation}
In other words, $U_\tau$ becomes an affine open subset of $U_\sigma$
when $\tau$ is a face of $\sigma$.  This will be useful in
Chapters~\ref{poly} and \ref{normal} when we study the local structure
of more general toric varieties.

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

\begin{example}
\label{1.latticechangeex}
In Example~\ref{1.rncnormal} the dual of $\sigma = \Cone(e_1,e_2)
\subseteq \R^2$ interacts with the lattices shown in Figure~10 on the
next page.
\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.8in]{ch1fig10.ps}
\end{center}
\caption{Lattice points of $\sigma^\vee$ relative to two lattices}  
\end{figure}
To make this precise, let us name the lattices involved:\ 
the lattices
\[
N' = \Z^2 \subseteq N = \{(a/2,b/2) \mid a,b \in \Z,\ a+b \equiv 0
\bmod 2\}
\]
have $\sigma \subseteq N'_\R \subseteq \nr$, and the dual lattices
\[
M' = \Z^2 \supseteq M =  \{(a,b) \mid a,b \in \Z,\ a+b \equiv 0
\bmod 2\}
\]
have $\sigma^\vee \subseteq M'_\R \subseteq \mr$.  Note that duality
reverses inclusions and that $M$ and $N$ are indeed dual under dot
product.  In Figure~10 (a), the black dots in the first quadrant form
the semigroup $\sS_{\sigma,N'} = \sigma^\vee\cap M'$, and in Figure~10
(b), the white dots in the first quadrant form $\sS_{\sigma,N} =
\sigma^\vee\cap M$.

This gives the affine toric varieties $U_{\sigma,N'}$ and
$U_{\sigma,N}$.  Clearly $U_{\sigma,N'} = \C^2$ since $\sigma$ is
smooth for $N'$, while Example~\ref{1.rncnormal} shows that
$U_{\sigma,N}$ is the rational normal cone $\widehat{C}_2$.
The inclusion $N' \subset N$ gives a toric morphism
\[
\C^2 = U_{\sigma,N'} \longrightarrow U_{\sigma,N} = \widehat{C}_2.
\]
Our next task is to find a nice description of this map.\eoe
\end{example}

In general, suppose we have lattices $N' \subseteq N$, where $N'$ has
finite index in $N$, and let $\sigma \subseteq N'_\R = \nr$ be a
strongly convex rational polyhedral cone.  Then the inclusion $N'
\subseteq N$ gives the toric morphism
\[
\varphi : U_{\sigma,N'} \longrightarrow U_{\sigma,N}.
\]
The dual lattices satisfy $M' \supseteq M$, so that $\varphi$
corresponds to the inclusion
\[
\C[\sS_{\sigma,N'}] = \C[\sigma^\vee\cap M'] \supseteq
\C[\sigma^\vee\cap M] = \C[\sS_{\sigma,N}]
\] 
of semigroup algebras.  The idea is to realize $\C[\sS_{\sigma,N}]$ as
a ring of invariants of a group action on $\C[\sS_{\sigma,N'}]$.  The
group in question is the finite abelian group
\begin{equation}
\label{1.gdef}
G = \mathrm{Hom}_\Z(M'/M,\Cstar),
\end{equation}
which acts on  $\C[\sS_{\sigma,N'}]$ via
\[
g \cdot \chi^{m'} = g([m'])\, \chi^{m'}.
\]
One can prove that $m' \in M'$ lies in $M$ if and only if $g([m']) =
1$ for all $g \in G$.  Hence the ring of invariants
\[
\C[\sS_{\sigma,N'}]^G = \{f \in \C[\sS_{\sigma,N'}] \mid g\cdot f = f
\text{ for all } g \in G\},
\]
is precisely $\C[\sS_{\sigma,N}]$, i.e.,
\[
\C[\sS_{\sigma,N'}]^G = \C[\sS_{\sigma,N}].
\]
In terms of the corresponding affine toric varieties, this has the
following geometric consequence.

\begin{proposition}
\label{1.quotient}
If $N'$ has finite index in $N$ and $\sigma \subseteq N'_\R = \nr$ is
a strongly convex rational polyhedral cone, then the finite group $G$
defined in \eqref{1.gdef} acts on $U_{\sigma,N'}$ and the toric
morphism $U_{\sigma,N'} \to U_{\sigma,N}$ induces a bijection
\[
U_{\sigma,N'}/G \simeq U_{\sigma,N}.
\]
\end{proposition}

\begin{proof}
When a finite group $G$ acts on an affine variety $V = \Spec(R)$ over
$\C$, we get a set $V/G$ of $G$-orbits as well as an
action of $G$ on the coordinate ring $R$.  The key fact is that
$V/G$ has a natural structure as an affine variety, namely 
\[
V/G \simeq \Spec(R^G),
\]
where $R^G$ is the ring of invariants of the $G$-action on $R$.  When
$V = \C^n$ and $G$ acts linearly, this is proved in \cite[Theorem 10
of Chapter 7]{iva}.  This proof adapts easily to the case when $R =
\C[\sS_{\sigma,N'}]$ (Exercise~\ref{1.3.quotientexer}).
\end{proof}

Here is an example of Proposition~\ref{1.quotient}.

\begin{example}
\label{1.redornc2ex}
In the situation of Example~\ref{1.latticechangeex}, one computes that
$G$ is the group $\mu_2 = \{\pm1\}$ acting on $U_{\sigma,N'} =
\Spec(\C[s,t]) \simeq \C^2$ by $-1\cdot(s,t) = (-s,-t)$.  Thus the
rational normal cone $\widehat{C}_2$ is the quotient
\[
\C^2/\mu_2 = U_{\sigma,N'}/\mu_2 \simeq U_{\sigma,N} = \widehat{C}_2.
\]
We can see this explicitly as follows.  The invariant ring is
easily seen to be
\[
\C[s,t]^{\mu_2} = \C[s^2,st,t^2] = \C[\widehat{C}_2] \simeq
\C[x_0,x_1,x_2]/\langle x_0x_2-x_1^2 \rangle,
\]
where the last isomorphism follows from Example~\ref{1.rnccone}.  From
the point of view of invariant theory, the generators $s^2,st,t^2$ of
the ring of invariants give a morphism
\[
\Phi : \C^2 \longrightarrow \C^3,\quad (s,t) \to (s^2,st,t^2)
\]
that is constant on $\mu_2$-orbits.  This map also separates orbits,
so it induces 
\[
\C^2/\mu_2 \simeq \Phi(\C^2) = \widehat{C}_2,
\]
where the last equality is by Example~\ref{1.rnccone}.  But we can
also think about this in terms of semigroups, where the exponent
vectors of $s^2,st,t^2$ give the Hilbert basis of the semigroup
$\sS_{\sigma,N}$ pictured in Figure~11 (b).  Everything fits together
very nicely. \eoe
\end{example}

In Exercise~\ref{1.3.rncexer} you will generalize
Example~\ref{1.redornc2ex} to the case of the rational normal
cone $\widehat{C}_d$ for arbitrary $d$.

The group $G$ from \eqref{1.gdef} has some interesting properties
(Exercise~\ref{1.3.giskernel}).

\begin{proposition}
\label{1.gprops}
Let $N$ be a lattice and $N' \subseteq N$ a sublattice of finite
index, with dual lattices $M' \supseteq M$.  Define $G =
\mathrm{Hom}_\Z(M'/M,\Cstar)$ as in \eqref{1.gdef}.  Then:
\begin{thmenum}
\item There is a perfect bilinear pairing $N/N' \times M'/M \to
  \Cstar$ that gives a canonical isomorphism $N/N' \simeq G$.
\item $G$ is the kernel of the map of tori $T_{N'} \to T_N$
induced by $N' \subseteq N$.
\end{thmenum}
\end{proposition}

Here is our final example of Proposition~\ref{1.quotient}.

\begin{example}
\label{1.simplquot}
Let $\sigma \subset \nr \simeq \R^n$ be a simplicial cone of dimension
$n$ with ray generators $u_1,\dots,u_n$.  Then $N' = \sum_{i=1}^n \Z
u_i$ is a sublattice of finite index.  Furthermore, $\sigma$ is smooth
relative to $N'$, so that $U_{\sigma,N'} = \C^n$.  It follows that $G
= N/N'$ acts on $\C^n$ with quotient
\[
\C^n/G = U_{\sigma,N'}/G \simeq U_{\sigma,N}.
\]
Hence the affine toric variety of a simplicial cone is the quotient of
affine space by a finite abelian group.  In the literature, varieties
like $U_{\sigma,N}$ are called \emph{orbifolds} and are said to be
\emph{$\Q$-factorial}.
\end{example}

\exercises

\small

\begin{exercise}
\label{1.3.1} 
Consider the affine toric variety $\ya = \Spec(\C[\sS])$, where $\cA =
\{m_1,\dots,m_s\}$ and $\sS = \N\cA$.  Let $\gamma : \sS \to \C$ be a
semigroup homomorphism.  In the proof of Proposition~\ref{1.points} we
showed that $p = (\gamma(m_1),\dots,\gamma(m_s))$ lies in $\ya$.
\begin{thmenum}
\item Prove that the maximal ideal $\{f \in \C[\sS] \mid f(p) = 0\}$
  is the kernel of the $\C$-algebra homomophism $\C[\sS] \to \C$
  induced by $\gamma$.
\item The torus $T_N$ of $\ya$ has character lattice $M = \Z\cA$ and
  fix $t \in T_N$.  As in the discussion following
  Proposition~\ref{1.points}, this gives the semigroup homomorphism $m
  \mapsto \chi^m(t) \gamma(m)$.  Prove that this corresponds to the
  point 
\[
(\chi^{m_1},\dots,\chi^{m_s})\cdot (\gamma(m_1),\dots,\gamma(m_s)) =
(\chi^{m_1}\gamma(m_1),\dots,\chi^{m_s}\gamma(m_s))
\]
  coming from the action of $t \in T_N \subseteq (\Cstar)^s$ on $p \in
  \ya \subseteq \C^s$.
\end{thmenum} 
\end{exercise}

\begin{exercise}
\label{1.3.torusaction}
Let $V = \Spec(\C[\sS])$ with for torus $T_N = \Spec(\C[M])$, $M =
\Z\sS$.  The action $T_N \times V \to V$ comes from a $\C$-algebra
homomorphism $\C[\sS] \to \C[M]\otimes_\C \C[\sS]$.  Prove that this
homomorphism is given by $\chi^m \mapsto \chi^m\otimes\chi^m$.  Hint:\
First show that this formula determines the $\C$-algebra
homomorphism $\C[M] \to \C[M]\otimes_\C \C[M]$ that gives the group
operation $T_N \times T_N \to T_N$.
\end{exercise}

\begin{exercise}
\label{1.3.1a} 
Prove Corollary~\ref{1.fixptcone}.
\end{exercise}

\begin{exercise}
\label{1.3.2} 
Let $\cA \subseteq M$ be a finite set.
\begin{thmenum}
\item Prove that the semigroup $\N\cA$ is saturated in $M$ if
  and only if $\N\cA = \Cone(\cA) \cap M$.  Hint:\ Apply
  \eqref{1.coneoverQ} to $\Cone(\cA) \subseteq \mr$.
\item Complete the proof of (b) $\Rightarrow$ (c) from
  Theorem~\ref{1.satthm}.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{1.3.3} 
Let $N$ be a lattice.  
\begin{thmenum}
\item Let $N_1 \subseteq N$ be a sublattice such that $N/N_1$ is
  torsion-free.  Prove that there is a sublattice $N_2 \subseteq N$ such
  that $N = N_1\oplus N_2$.
\item Let $u \in N$ be primitive as defined in the proof of
  Theorem~\ref{1.satthm}.  Prove that $N$ has a basis $e_1,\dots,e_n$
  such that $e_1 = u$.  
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{1.3.4}
Prove Proposition~\ref{1.normalize}.
\end{exercise}

\begin{exercise}
\label{1.3.5}
Let $p$ be a point of an irreducible affine variety $V$.  Then $p$
gives the maximal ideal $\mf = \{f \in \C[V] \mid f(p) = 0\}$ as well
as the maximal ideal $\mf_{V,p} \subseteq \cO_{V,p}$ defined in \S0.
Prove that the natural map $\mf/\mf^2 \longrightarrow
\mf_{V,p}/\mf_{V,p}^2$ is an isomorphism of $\C$-vector spaces.
\end{exercise}

\begin{exercise}
\label{1.3.sm1}
Prove \eqref{1.secondshow}.  Hint:\ Use Lemma~\ref{1.computeZar} and
Example~\ref{1.firstproduct}.
\end{exercise}

\begin{exercise}
\label{1.3.conemorphismexer}
Prove Proposition~\ref{1.3.conemorphism}.
\end{exercise}

\begin{exercise}
\label{1.3.quotientexer}
Prove Proposition~\ref{1.quotient}.
\end{exercise}

\begin{exercise}
\label{1.3.rncexer}
The group of $d$th roots of unity $\mu_d = \{\zeta \in \Cstar \mid
\zeta^d = 1\}$ be acts on $\C^2$ by $\zeta\cdot(x,y) = (\zeta x, \zeta
y)$.  Adapt Example~\ref{1.redornc2ex} to show that $\C^2/\mu_d \simeq
\widehat{C}_d$.  Hint: Use lattices $N' = \Z^2 \subseteq N =
\{(a/d,b/d) \mid a,b \in \Z, a+b \equiv 0 \bmod d\}$.
\end{exercise}

\begin{exercise}
\label{1.3.giskernel}
Prove Proposition~\ref{1.gprops}.  Hint:\ Use the pairing $\langle \
,\ \rangle : M_\Q \times N_\Q \to \Q$ and note that $M \subseteq M'
\subseteq M_\Q$ and $N' \subseteq N \subseteq N_\Q$.  The map $\Q/\Z
\to \Cstar$ given by $[q] \mapsto e^{2\pi i q}$ will also be helpful.
\end{exercise}

\begin{exercise}
\label{1.3.prod}
Let $\sigma_1 \subset (N_1)_\R$ and $\sigma_2 \subset (N_2)_\R$ be
strongly convex rational polyhedral cones.  This gives the cone
$\sigma_1\times\sigma_2 \subset (N_1\oplus N_2)_\R$.  Prove that
$U_{\sigma_1\times\sigma_2} \simeq U_{\sigma_1}\times U_{\sigma_2}$.
Also explain how this result applies to \eqref{1.proddecomp}.
\end{exercise}

\begin{exercise}
\label{1.3.sgtorus}
By Proposition~\ref{1.points}, a point $p$ of an affine toric variety
$V = \Spec(\C[\sS])$ is represented by a semigroup homomorphism
$\gamma : \sS \to \C$.  Prove that $p$ lies in the torus of $V$ if and
only if $\gamma$ never vanishes, i.e., $\gamma(m) \ne 0$ for all $m
\in \sS$.
\end{exercise}

\normalsize

