\chapter{Normal Toric Varieties}
\label{normal}

\setcounter{section}{-1}

\section{Background:\ Abstract Varieties}

%\begin{itemize}
%    \item{abstract varieties, gluing}
%    \item{products of varieties}
%    \item{def. of separated}
%\end{itemize}

The projective toric varieties studied in Chapter~\ref{poly} are
unions of Zariski open affine pieces, each of which is an affine 
variety.  We will begin this chapter by building up to a general construction 
of abstract varieties obtained by gluing together affine varieties in
an analogous way.  The resulting varieties will be \emph{abstract} in
the sense that they do not come with any given ambient affine or 
projective space.  We will see that this is exactly the idea needed 
to construct a toric variety using the combinatorial data contained 
in a fan and to study the different ways such varieties may be
mapped into affine or projective spaces.  

To provide some motivation, we recall the definition of a differentiable 
manifold.  Roughly speaking, a manifold is a topological space with the 
property that every point has an open neighborhood that is homeomorphic 
to a subset of Euclidean space, and these coordinate neighborhoods fit 
together properly so that one can define differentiable functions on the 
whole space.  More precisely, let $X$ be a Hausdorff topological space with a 
countable basis.  Assume that $X$ is covered by open subsets $U_\a$ 
for which there exist homeomorphisms $\phi_\a : U_\a \to V_\a$
where $V_\a \subseteq \R^n$ is open for each $\a$. 
Assume that whenever $U_\a \cap U_\b \ne \emptyset$, 
$\phi_\b \circ \phi_\a^{-1}$ is a $C^m$-differentiable
mapping from $\phi_\a(U_\a \cap U_\b) \subseteq \R^n$ to 
$\phi_\b(U_\a \cap U_\b)\subseteq \R^n$.  Then we say 
$X$ has the structure of a $C^m$ differentiable $n$-manifold.  

The definition of an abstract variety is similar in spirit.
Roughly speaking again, an abstract variety is a topological 
space which has the property that every point has an open
neighborhood that has the structure of an affine algebraic variety, 
and these coordinate neighborhoods fit together properly so that 
one can define regular functions.  

\subsection*{\emph{Sheaves}}
Although it is possible to follow the definition of a manifold given
above to construct abstract varieties, we will present a somewhat
different approach based on the important concept of a sheaf.  Our
approach will be a special case of the theory of schemes presented
in \cite{eh} and \cite{hartshorne}.  The language of sheaves will
also be used in later chapters.  

\begin{definition}
\label{3.sheafdef}
Let $X$ be a topological space, and let $k$ be a field.  
A \emph{\bfseries sheaf} $\mathcal{F}$ of $k$-algebras on $X$ consists of the data
\begin{thmitem}
\item for each open set $U \subseteq X$, a $k$-algebra $\mathcal{F}(U)$, and
\item for each inclusion $V \subseteq U$ of open subsets of $X$, a
$k$-algebra homomorphism $\rho_{U,V} : \mathcal{F}(U) \to \mathcal{F}(V)$,
\end{thmitem} 
satisfying the following properties:
\begin{thmenum}
\item $\mathcal{F}(\emptyset) = \{0\}$.
\item For all open sets $U$, $\rho_{U,U}$ is the identity mapping.
\item For all nested sequences of open sets $W \subseteq V \subseteq U$, 
\[
\rho_{V,W}\circ\rho_{U,V}  = \rho_{U,W}.
\]
\item If $U = \bigcup_\a U_\a$ and $f \in \mathcal{F}(U)$
satisfies $\rho_{U,U_\a}(f) = 0$ for all $\a$, then $f = 0$.
\item If $U = \bigcup_\a U_\a$, and a collection of
$f_\a \in \mathcal{F}(U_\a)$ satisfies
\[
\rho_{U_\a,U_\a\cap U_\b}(f_\a) = \rho_{U_\b,U_\a\cap U_\b}(f_\b)
\]
for all $\a, \b$, then there exists $f\in \mathcal{F}(U)$ such
that $f_\a = \rho_{U,U_\a}(f)$ for all $\a$.
\end{thmenum}
The elements of $\mathcal{F}(U)$ are called the \emph{\bfseries sections} of 
$\mathcal{F}$ over the open set $U$, and $\rho_{U,V}$ is called the 
\emph{\bfseries restriction mapping} from sections over $U$ to sections over $V$.  
We say that the pair $(X,\mathcal{F})$ is a \emph{\bfseries ringed space over $k$}.  
\end{definition}

Modifying this definition according to the structure
required on the $\mathcal{F}(U)$, one can also 
also consider sheaves of abelian groups, modules, etc.
Items d and e are equivalent to the exactness of the sequence
\[
0 \to \mathcal{F}(U) \to \prod_{\a} \mathcal{F}(U_\a)\  
%\lower3pt\hbox{$\smash{\mathop{\to}\limits^{\displaystyle \to}}$}\
\Atop{\to}{\to}      
\prod_{\a,\b} \mathcal{F}(U_\a\cap U_\b),
\]
where the second mapping from the left is defined by the restrictions
$\rho_{U,U_\a}$ and the $\mathcal{F}(U_\a\cap U_\b)$-component 
of the mapping indicated by the double arrow is the 
difference of the restrictions $\rho_{U_\a,U_\a\cap U_\b}$
and $\rho_{U_\b,U_\a\cap U_\b}$.  Exactness at $\mathcal{F}(U)$
implies that sections of $\mathcal{F}$ are determined locally (that is,
two sections over $U$ are equal if their restrictions to all $U_\a$
are equal), which is equivalent to part d.
Similarly, exactness at $\prod_{\a} \mathcal{F}(U_\a)$ 
means that a family of sections over the $U_\a$ agreeing on all 
the overlaps $U_\a \cap U_\b$ patches together to give a section over 
$U$, which is part e.  

For example, if $X \subseteq \R^n$ and $k = \R$, then 
we obtain a sheaf $\mathcal{F} = \cO^m_X$ of $\R$-algebras on $X$ by
setting 
\[
\cO^m_X(U) = \{f : U \to \R\ \vert\ f \text{ is a } C^m \text{ function }\}.
\] 
so that $(X,\cO^m_X)$ is a ringed space over $\R$.  

We will also need a notion of \emph{morphisms} and 
\emph{isomorphisms} of ringed
spaces over $k$.  

\begin{definition}
\label{3.morphringsp}
An \emph{\bfseries morphism of ringed spaces} $(X,\mathcal{F})$ and
$(Y,\mathcal{G})$ consists of the data of
\begin{coxitem}
\item a continuous mapping $\phi : X \to Y$, and 
\item for each open set $U \subseteq Y$, an homomorphism of
$k$-algebras
\[
\phi^\#_U : \mathcal{G}(U) \to \mathcal{F}(\phi^{-1}U),
\]
such that whenever $V \subseteq U$ are open sets in $Y$, the diagram
\[
\begin{array}{ccc}
\mathcal{G}(U) & \longrightarrow & \mathcal{F}(\phi^{-1}(U))\\
\downarrow     &                 & \downarrow\\
\mathcal{G}(V) & \longrightarrow & \mathcal{F}(\phi^{-1}(V))
\end{array}
\]
commutes, where the vertical arrows are the corresponding restrictions,
and the horizontal arrows are $\phi^\#_V$ and $\phi^\#_U$, respectively.  
\end{coxitem}

If $\phi$ is a homeomorphism and the $\phi^\#_U$ are all
$k$-algebra isomorphisms, then $\phi$ is said to be an 
\emph{\bfseries isomorphism of ringed spaces} over $k$.
\end{definition}

We think of the sections of $\mathcal{F}$ and 
$\mathcal{G}$ as mappings so $\phi^\#_U$ 
takes $g\in \mathcal{G}(U)$ to $g\circ \phi$ in 
$\mathcal{F}(\phi^{-1}(U))$.

In Exercise~\ref{3.manifold}, you will show that the definition 
of a manifold can be reformulated using the language of sheaves of 
$\R$-algebras and isomorphism of ringed spaces.  

\subsection*{\emph{The Structure Sheaf of an Affine Variety}}
Let $V = \Spec(R)$ be an affine variety.
In \S 0 of Chapter 1, we defined the Zariski open subset
$V_f = V\ \setminus\ \V(f) \subseteq V$ for all $f \in R$.  
We also saw in Exercise~0.2 of that section that
$V_f = \Spec(R_f)$, where $R_f$ is the localization of $R$.  
The open sets $V_f$ form a \emph{basis} for the Zariski topology
on $V$ in the sense that every open set $U$ is a (finite) union
$U = \bigcup_{f\in S} V_f$ for some $S\subseteq R$ (Exercise~\ref{3.quasicompact}).

We now prepare for the definition of the structure sheaf of $V$,
a sheaf of $\C$-algebras on $V$ with the Zariski topology.  

\begin{definition}
\label{3.regfundef}
Given an affine variety $V = \Spec(R)$ and a 
Zariski open $U \subseteq V$, we say $\phi : U \to \C$
is \emph{\bfseries regular on} $U$ if for all $p \in U$,
there exists an $f_p \in R$ such that $p \in V_{f_p} \subseteq U$
and $\phi |_{V_{f_p}} \in R_{f_p}$.  
\end{definition}

Note that the condition $p \in V_{f_p}$ means that $f_p(p) \ne 0$, 
and saying $\phi |_{V_{f_p}} \in R_{f_p}$ means that $\phi = a_p/f_p^{n_p}$
for some $a_p \in R$ and $n_p \ge 0$.  

The structure sheaf on $V$ is the sheaf $\cO_V$ whose sections
over the open set $U$ are defined to be
\[
\cO_V(U) = \{ \phi : U \to \C\ |\ \phi \text{ is regular }\}.
\]
When $W \subseteq U$, Definition~\ref{3.regfundef}
shows that there is an obvious restriction mapping 
$\rho_{U,W} : \cO_V(U) \to \cO_V(W)$.  The following proposition 
gives some first properties of structure sheaves. 

\begin{proposition}
\label{3.strucsheafprops}
Let $V = \Spec(R)$ be an irreducible affine variety.
\begin{coxenum}
\item $\cO_V(V) = R$.
\item If $f \in R$, then $\cO_V(V_f) = R_f$.
\end{coxenum}
\end{proposition}

\begin{proof}
It is clear from Definition~\ref{3.regfundef} that each element of $R$ 
defines a regular function on $V$, hence a section in $\cO_V(V)$.  
Conversely, if $\phi \in \cO_V(V)$, then for all $p \in V$ we can 
find $f_p \in R$ such that $p \in V_{f_p}$
and $\phi = a_p/f_p^{n_p}\in R_{f_p}$.  The ideal 
$I = \langle f_p^{n_p}\ |\ p \in V\rangle \subseteq R$ 
satisfies $\V(I) = \emptyset$ since $f_p(p) \ne 0$  for all $p\in V$.  
Hence the Nullstellensatz implies that $\sqrt{I} = \I(\V(I)) = R$, 
so there exists a finite set $S\subseteq V$ and polynomials $g_p$ 
for $p \in S$ such that 
\[
1 = \sum_{p\in S} g_p f_p^{n_p}.
\]
Hence, 
\[
\phi = \sum_{p\in S} g_p f_p^{n_p}\phi = \sum_{p\in S} g_p a_p,
\]
which shows that $\phi \in R$.  

For part b, we use the fact that $V_f = \Spec(R_f)$ is an affine 
variety with a structure sheaf of its own.
To show $\cO_V(V_f) = R_f$, it suffices to show that if 
$U \subseteq V_f$ is Zariski open, then regarding $U$ as a subset
of $V$, $\cO_V(U) = \cO_{V_f}(U)$.  Then taking $U = V_f$ we
obtain the desired result, by part a.  Because we have $R \hookrightarrow R_f$
and $R_{f_p} \hookrightarrow (R_{f_p})_f$ when $R$ is a domain,
$\cO_V(U) \subseteq \cO_{V_f}(U)$ is immediate.  Conversely, 
let $\phi \in \cO_{V_f}(U)$.  Then for all $p \in U$, there
is some $f_p \in R_f$ such that $p \in V_{f_p} \subseteq U \subseteq V_f$
and $\phi = a_p/f_p^{n_p}$.  Since $f_p \in R_f$,
we can write $f_p = g_p/f^{m_p}$.  Since $f_p(p) \ne 0$, it must be 
true that $g_p(p) \ne 0$ also.  Then $\phi = (a_p f^{m_p n_p})/g_p^{n_p}$.
Hence for all $p \in U$, there exist $g_p$
such that $g_p(p) \ne 0$ and $\varphi \in R_{g_p}$, which
shows that $\phi \in \cO_V(U)$.  
\end{proof} 

The first part of Proposition~\ref{3.strucsheafprops}
can also be restated using the \emph{local rings} $\cO_{V,p}$
introduced in Chapter~\ref{affine}, \S 0.  A rational function
$f/g \in \C(V)$ is contained in the local ring $\cO_{V,p}$
precisely when it defines a section of $\cO_V$ over 
some open set $U$ containing $p$.  Hence
\begin{equation}
\label{3.intlocalrings}
\bigcap_{p\in V} \cO_{V,p} = \cO_V(V) = R = \C[V].
\end{equation}

We have not, of course, verified that the definition 
given above actually makes $\cO_V$ into a sheaf (we have not shown
that all the properties in Definition~\ref{3.sheafdef}
hold).  In fact, one standard method here is to use the 
fact that the $V_f$ give a basis for the Zariski topology on $V$.
One can take the statement of part b of Proposition~\ref{3.strucsheafprops}
as the \emph{definition} of $\cO_V(V_f)$ for the special open sets $V_f$.
One verifies that the sheaf properties hold for open
sets of this form (you will do this in Exercise~\ref{3.prinaffineopens}).
Then one shows that there exists a unique way to extend the 
definition of $\cO_V(U)$ to arbitrary open sets $U$ to satisfy
the sheaf properties.  See \cite{eh} for this approach.    

\begin{example}
\label{3.strucsheaf}
Let $V = \C^2 = \Spec(\C[x,y])$ and let
$U = V_x = V\ \setminus\ \V(x)$ and $W = V\ \setminus\ \{(0,0)\}$
be two Zariski open subsets of $V$.   By part b of 
Proposition~\ref{3.strucsheafprops}, $\cO_V(U)$ is the 
localization $\C[x,y]_x$.  

We claim next that $\cO_V(W) = \C[x,y]$.  This follows
since $W = V_x \cup V_y$.  By part e of Definition~\ref{3.sheafdef},
and part b of the proposition, we have 
\[
\cO_V(W) = \cO_V(V_x) \cap \cO_V(V_y) = \C[x,y]_x \cap \C[x,y]_y = \C[x,y].
\]
The restriction mapping $\rho_{W,U}$ in this
case coincides with the standard inclusion 
$\C[x,y] \hookrightarrow \C[x,y]_x$.
\eoe
\end{example}

\subsection*{\emph{The Definition of an Abstract Variety}}
We now have all of the ingredients to define abstract 
varieties. Note the analogy with the definition of a differentiable 
manifold mentioned at the start of this section as restated in 
Exercise~0.1.

\begin{definition}
\label{3.absvardef}
An \emph{\bfseries abstract variety} is a ringed space $(X,\cO_X)$
over $\C$ with the property that every $p\in X$ has a neighborhood 
$U$ such that the restriction $(U,\cO_X|_U)$ is isomorphic as a ringed
space over $\C$ to $(V,\cO_V)$ for some affine variety $V$.
\end{definition}

An abstract variety $(X,\cO_X)$ comes equipped with the Zariski topology
whose open sets are those sets that
restrict to open sets in each $(U,\cO_X|_U)$
isomorphic to the ringed space of an affine variety.  The Zariski 
closed subsets $Y \subseteq X$ are called \emph{subvarieties} of $X$.
Any such $Y$ inherits the structure of an abstract variety
with structure sheaf $\cO_Y$ defined as follows.  If $U\subseteq Y$
is open, then we set $\cO_Y(U)$ to be the set of all $\phi : U \to \C$
such that for all $p\in U$, there exist $U_1$ open in $X$ and
and a regular function $\phi_1 : U_1 \to \C$ with 
$p \in U_1\cap Y \subseteq U$ and $\phi|_{U_1 \cap Y} = \phi_1|_{U_1 \cap Y}$.
You will show that this defines the structure sheaf
of $Y$ and makes $Y$ an abstract variety in Exercise~\ref{3.subvariety}.  

As noted in Chapter~\ref{affine}, affine varieties $V \subseteq \C^n$ also 
have a \emph{classical} topology.  It follows that abstract 
varieties have a \emph{classical} topology too, the smallest
collection of open sets that agrees with the 
classical topology on every affine open subset.  
Since we work over $\C$, there is also a sheaf $\cO_V^{an}$ whose sections 
over an open set $U$ in the classical topology are complex analytic functions 
$\phi : U \to \C$.  The relation between the variety $V$ and the associated 
\emph{complex analytic space} $(V^{an},\cO_V^{an})$ is studied in \cite{serre} and more 
briefly in Appendix B of \cite{hartshorne}.

An abstract variety $(X,\cO_X)$ is said to be \emph{irreducible} if it is not
the union of two proper subvarieties.  This is true if and only if
$X$ is connected and each of its affine open subsets is irreducible as
an ordinary affine variety.  

\begin{example}
\label{3.absproj}
The usual description of $\P^n$ via homogeneous coordinates \emph{is} 
a construction of $\P^n$ as an abstract variety.  Moreover every projective
variety $V \subseteq \P^n$ is also an abstract variety.
Indeed, as in Equation~\eqref{2.uimap} from Chapter~\ref{poly}, for $i = 1, \ldots, n$
we have the affine open sets
\[
U_i = \P^n\ \setminus\ \V(x_i) \simeq \Spec(\C[x_1/x_i,\ldots,x_{i-1}/x_i,x_{i+1}/x_i,\ldots,x_n/x_i])
\]
that cover $\P^n$.  And similarly the affine opens $V_i = V \cap U_i$ cover $V$.

The only new element here is the structure
sheaf.  First, we note that the set of quotients $f/g$ with 
$f,g \in \C[x_0,\ldots,x_n]$ and $f,g$ homogeneous of the same degree
forms a field, called the \emph{function field} of $\P^n$ and written $\C(\P^n)$.
Given an open set $U \in \P^n$, we say $\phi : U \to \C$ is a regular function  
if for all $p\in U$ there is some $f/g \in \C(\P^n)$ such that $g(p) \ne 0$
and $\phi = f/g$ when restricted to $U\ \setminus\ U \cap \V(g)$.  We then define the 
sheaf $\cO_{\P^n}$ by assigning
\[
\cO_{\P^n}(U) = \{ \phi: U \to \C\ \vert\ \phi \text{ is regular}\}.
\]

If $f,g$ are homogeneous of degree $d$, then rewriting
$f/g = (f/x_i^d)/(g/x_i^d)$ shows that there is an isomorphism 
\begin{equation}
\label{3.funfield}
\C(\P^n) \simeq \C(x_1/x_i,\ldots,x_{i-1}/x_i,x_{i+1}/x_i,\ldots,x_n/x_i)
\end{equation}
for each $i$.  Using these isomorphisms, it is not difficult to show
that $(U_i,\cO_{P^n}|_{U_i})$ is isomorphic to $(\C^n,\cO_{\C^n})$ as
a ringed space over $\C$.  Note that $\P^n$ is irreducible as an abstract
variety.

The fact that a projective variety $V \subseteq \P^n$ is an
abstract variety now follows from Exercise~0.5.

On $\P^n$ there are very few
\emph{global sections} of the structure sheaf (that is, sections over $U = \P^n$).
In fact, in Exercise~\ref{3.globalsecs}, you will show that $\cO_{\P^n}(\P^n) = \C$.  Compare
this with part a of Proposition~\ref{3.strucsheafprops} in the affine case.
\eoe
\end{example}

{}From now on, when we consider varieties, we will usually
be thinking of abstract varieties.  For brevity, 
we will often say ``the variety $X$'' rather than ``the abstract 
variety $(X,\cO_X)$''.

\subsection*{\emph{Function Field and Local Rings}}
The function field $\C(\P^n)$ introduced in \eqref{3.funfield}
has a counterpart for any irreducible variety $X$.  Namely, we can 
define a \emph{rational function} on $X$ to be a regular function 
$f : U \to \C$ where $U$ is some nonempty Zariski open set in $X$.  
We say two rational functions on $X$ are equivalent if they agree on 
some nonempty Zariski open subset.  In Exercise~\ref{3.funcfieldprops}, 
you will show that this relation is an equivalence relation, and the set 
of equivalence classes has the structure of a field, called the function 
field of $X$ and denoted $\C(X)$.

We say that a rational function $f \in \C(X)$ is \emph{defined at} $p \in X$
if $f$ is equivalent to a regular function $g : U \to \C$ such that
$p \in U$.

\begin{definition}
\label{3.localringdef}
Let $X$ be an irreducible variety and $p\in X$.
The \emph{\bfseries local ring of $X$ at $p$} is
\[
\cO_{X,p} = \{ \phi \in \C(X)\ \vert\ \phi \text{ is defined at } p\}.   
\]
\end{definition}

It is not difficult to see that $\cO_{X,p}$ is a subring
of $\C(X)$ satisfying the same properties as in the affine case.

\begin{proposition}
\label{3.localringprop}
The ring $\cO_{X,p}$ is a local ring with unique maximal
ideal 
\[
\mf_{X,p} = \{\phi \in \cO_{X,p}\ |\ \phi(p) = 0\}.  
\]
\end{proposition}

The local ring $\cO_{X,p}$ can also be defined for general
(possibly reducible) varieties as the direct limit
\[
\cO_{X,p} = 
\lim_{{\scriptstyle \longrightarrow} \atop {\scriptstyle p\in U}} 
\cO_X(U), 
\]
over all neighborhoods of $p$ in $X$.  See \cite{eh} or 
\cite{hartshorne}.

\subsection*{\emph{Normal Varieties}}
We now extend the notion of normality of varieties introduced
in Chapter~\ref{affine}.

\begin{definition}
\label{3.normalvardef}
We say that an irreducible variety $X$ is 
\emph{\bfseries normal} if the local ring $\cO_{X,p}$ is
integrally closed for all $p \in X$.
\end{definition}

At first glance, this appears to be 
different from the definition given for affine
varieties in Chapter~\ref{affine}, Definition~\ref{1.normaldef}.  
To reconcile the two notions, we need to recall the
localization $R_S$ of a domain $R$ at a \emph{multiplicative
subset} $S$ (that is a subset that is closed under multiplication
with $1\in S, 0\notin S$).  The construction is the same as in
the special case $S = \{f^n\ \vert\ n\ge 0\}$ used to produce
the ring $R_f$.  Elements of $R_S$ are fractions $a/s$ with
$a \in R$ and $s\in S$, and $a/s = a'/s'$ if $s'a = sa'$ in $R$.
In the affine case, the local ring $\cO_{X,p}$ is just the 
localization $\C[X]_{S_p}$ of $\C[X]$ at the multiplicative subset
$S_p = \C[X]\ \setminus\ \I(\{p\}) = \{f \in \C[X]\ \vert\ f(p) \ne 0\}$.

It is a standard result in commutative algebra that if $R$ is an
integrally closed domain and $S$ is any multiplicative subset, then
$R_S$ is also integrally closed (see \cite{eisenbud}, 4.13, for instance).  
Hence the definition we gave in Chapter~\ref{affine}
implies the condition in Definition~\ref{3.normalvardef}.
In fact the two conditions are equivalent for affine varieties.

\begin{proposition}
\label{3.normaldefequiv}
Let $V$ be an irreducible affine variety.  Then $\C[V]$ is 
integrally closed if and only if the local ring $\cO_{V,p}$
is integrally closed for all $p \in V$.
\end{proposition}

\begin{proof}
It suffices to show that if $\cO_{V,p}$ is integrally 
closed for all $p$, then $\C[V]$ is integrally closed.
But this follows from \eqref{3.intlocalrings} 
and the fact that any intersection of integrally
closed domains with the same field of fractions 
is integrally closed (Chapter~\ref{affine}, Exercise~0.5).  
\end{proof}

We also have the following direct consequence of 
Proposition~\ref{3.normaldefequiv} and Definition~\ref{3.normalvardef}.

\begin{proposition}
\label{3.normalcover}
Let $X$ be an irreducible variety with a cover consisting
of affine open sets $V_\a$. Then $X$ is normal if and only
if each $V_\a$ is normal.
\end{proposition}

\subsection*{\emph{Morphisms}}  
In \S 0 of Chapter~\ref{affine}, we stated that giving 
a polynomial mapping $F : V_1 \to V_2$ between
affine varieties is the same as giving the $\C$-algebra
homomorphism $F^* : \C[V_2] \to \C[V_1]$ where
$F^*(g) = g \circ F$ for $g \in \C[V_2]$.  The corresponding
mappings for (abstract) varieties are called 
\emph{morphisms} $\phi: X \to Y$.  

\begin{definition}
\label{3.morphdef}
A \emph{\bfseries morphism} from a variety $X$ to a variety $Y$ 
is a morphism $(\phi, \phi^\#)$ of ringed spaces over $\C$ 
as in Definition~\ref{3.morphringsp} with the additional
property that for all $p\in X$, the map of local rings
$\phi^\#_p : \cO_{Y,\phi(p)} \to \cO_{X,p}$ induced
by $\phi^\#$ satisfies 
\[
(\phi^\#_p)^{-1}(\mf_{X,p}) = \mf_{Y,\phi(p)}.
\]
\end{definition}

The additional property in this definition is summarized
by saying that $\phi^\#_p$ is a \emph{local homomorphism},
and $(\phi,\phi^\#)$ satisfying this property are sometimes
called \emph{morphisms of locally ringed spaces} over $\C$.

\subsection*{\emph{Gluing Together Affine Varieties}}
We now come to the main way (abstract) varieties
will be used in the study of toric varieties.  Namely,
we want to be able to say that spaces that are constructed
by ``gluing together'' several affine varieties in a 
suitable way have the structure of varieties.  

An immediate consequence of Definition~\ref{3.absvardef} 
is that, in a certain sense, any variety can be constructed 
by gluing together the affine varieties isomorphic to the 
subsets in an affine open cover.  The discussion of $\P^n$ in 
Example~\ref{3.absproj} is a first example.  
To be more precise about exactly how this works, let 
$X = \bigcup_\a U_\a$ where $U_\a$ are Zariski open and 
for each $\scriptstyle{\a}$,  assume we have an isomorphism
$\phi_\a : U_\a \simeq V_\a$ 
where $V_\a$ is affine.  For each pair $\scriptstyle{\a,\b}$ we have 
the subset
$V_{\a\b} = \phi_\a(U_\a\cap U_\b) \subseteq V_\a$
and 
$V_{\b\a} = \phi_\b(U_\a\cap U_\b) \subseteq V_\b$.
The points in $U_\a\cap U_\b$ are the same in both cases, so 
the map $g_{\a\b} = \phi_\b \circ \phi_\a^{-1}$
defines an isomorphism from $V_{\a\b}$ to $V_{\b\a}$.
Furthermore, these mappings satisfy the following 
\emph{compatibility conditions} on intersections:
\begin{coxitem}
\item For all pairs of indices $\scriptstyle{\a, \b}$, we have
$g_{\b\a} = g_{\a\b}^{-1}$.
\item For all triples of indices $\scriptstyle{\a,\b,\g}$, we have
$g_{\a\b}(V_{\a\b}\cap V_{\a\g}) = V_{\b\a}\cap V_{\b\g}$ and
$g_{\a\g} = g_{\b\g}\circ g_{\a\b}$,
on $V_{\a\b}\cap V_{\a\g}$.  
\end{coxitem}

Now we turn this observation around.  Suppose we start from the 
data of a collection $\{V_\a\}_\a$ of affine varieties and suppose
that for all pairs of indices $\scriptstyle{\a,\b}$
we have Zariski open sets $V_{\a\b} \subseteq V_\a$, and isomorphisms
$g_{\a\b} : V_{\a\b} \simeq V_{\b\a}$ satisfying
the compatibility conditions above.  Then we claim that the following
process produces a variety $X$ covered by Zariski open subset $U_\a$
isomorphic to the $V_\a$, and in which the points in $V_{\a\b}$
and $V_{\b\a}$ have been identified (that is, the $V_\a$ have
been glued together along the $V_{\a\b}$).

To get the underlying topological space, we let $Y$ be the
disjoint union of the $V_\a$.  Define a relation $\sim$ 
on $Y$ by $a \sim b$ if and only if $a \in V_\a$, $b \in V_\b$
for some $\a,\b$, and $b = g_{\a\b}(a)$.  The first 
compatibility condition shows
that $\sim$ is reflexive and symmetric; the second shows that $\sim$
is transitive.  Hence $\sim$ is an equivalence relation and we
can form the quotient space $X = Y/\sim$  with the quotient
topology.  It can then be checked that $X$ is covered by affine 
open subset $U_\a \simeq V_\a$ and that the structure sheaves
$\cO_{V_\a}$ patch together to give a sheaf $\cO_X$, so that 
$(X,\cO_X)$ is a variety.  

Rather than proving these assertions in detail, we refer
the reader to \cite{eh} or \cite{hartshorne}.  We will present
a few examples to give some of the flavor of the construction.

\begin{example}
\label{3.projline}
Let $V_0 = \Spec(\C[u]) \simeq \C$ and  $V_1 = \Spec(\C[v]) \simeq \C$.  Let
$V_{01} = \Spec(\C[u]_u) \simeq \Cstar$ and $V_{10} = \Spec(\C[v]_v) \simeq \Cstar$.
Take the gluing map $g_{01} : V_{01} \to V_{10}$ corresponding to the map 
of $\C$-algebras defined by $v \mapsto u^{-1}$.  This is clearly an 
isomorphism.  The compatibility conditions are satisfied trivially
since there are only two $V_i$.  So following the general construction, 
as a set, the variety $X$ is the union of one copy of $\Cstar$ and two 
other points corresponding to $0 \in V_{01}$ and $0 \in V_{10}$. We claim the 
resulting variety is the projective line $\P^1$.  The standard affine opens
$U_0 = \Spec(\C[x_1/x_0])$ and $U_1 = \Spec(\C[x_0/x_1])$ in $\P^1$ 
are isomorphic to $V_0$ and $V_1$ as above
via $u \mapsto x_1/x_0$ and $v \mapsto x_0/x_1$, and they are glued together
in the same way.
\eoe
\end{example}

\begin{example}
\label{3.blowuppt}
Let $V_0 = V_1 = \C^2$, with $V_0 = \Spec(\C[u,v])$ and $V_1 = \Spec(\C[w,z])$.
In $V_0$, let $V_{01} = V_0\ \setminus\ \V(v)$, so $V_{01} = \Spec(\C[u,v]_v)$.
Similarly, let $V_{10} = V_1\ \setminus\ \V(z) = \Spec(\C[w,z]_z)$.  Let
$g_{01} : V_{01} \to V_{10}$ be the morphism of varieties corresponding
to the $\C$-algebra homomorphism $\C[w,z]_z \to \C[u,v]_v$ defined by 
$w \mapsto uv$ and $z \mapsto 1/v$.  Then $g_{01}$ is an isomorphism because
$g_{10}$ defined by $v \mapsto 1/z$ and $u \mapsto wz$ is its inverse.
As in Example~\ref{3.projline}, the compatibility conditions are satisfied.
The resulting variety $X$ can be identified with a subvariety $W$ of the product
$\C^2 \times \P^1$ with coordinates $(x,y)$ on the $\C^2$ and homogeneous 
coordinates $(x_0,x_1)$ on the $\P^1$. Indeed, 
take $W$ defined by the equation $xx_1 = yx_0$.  
Note that $W$ is covered by two affine opens $W_0 \simeq \Spec(\C[x,x_1/x_0])$ 
(since $y = x x_1/x_0$ when $x_0 \ne 0$) and 
$W_1 \simeq \Spec(\C[y,x_0/x_1])$ (since $x = yx_0/x_1$ when $x_1 \ne 0$).
These affine subsets are isomorphic
to $V_0$ and $V_1$ respectively and are glued together in
$W$ in exactly the same way that $V_0$ and $V_1$ are glued together in $X$.  
In this representation, we see that the fibers of the morphism
$W \to \C^2$ defined by the projection
$\C^2 \times \P^1 \to \C^2$ contain just one point for each $(x,y) \ne (0,0)\in \C^2$,
but the fiber over $(0,0) \in \C^2$ is a copy of $\P^1$.  
Intuitively, the points on this $\P^1$ are in one-to-one correspondence
with the lines through $(0,0) \in \C^2$ (since $y/x = x_1/x_0$).
The variety $X$ (or equivalently $W$) is called the \emph{blow-up of 
$\C^2$ at the point $(0,0)$}.  
\eoe
\end{example}

We will see later that both of these examples, as well as the one
in Exercise~\ref{3.projplane} can be realized as normal toric varieties.

\subsection*{\emph{Products of Varieties}}
As another example of abstract varieties and gluing, we indicate 
why the product $X\times Y$ of varieties $X$ and $Y$
also has the structure of a variety.  There
is a slight subtlety here because the topology on $X\times Y$ we want is not just the 
product topology.  For example, consider $\C^2 \simeq \C \times \C$.
By definition, a basis for the product topology consists of sets
$U_1 \times U_2$ where $U_i$ are Zariski open in $\C$.  The
complement of one of these sets is a union of collections
of ``horizontal'' and ``vertical'' lines in $\C^2$, so it 
is clear that a Zariski closed set in $\C^2$ such as $\V(y - x^2)$ 
cannot be obtained as an intersection of complements of the 
product basic open sets.  

So we start by considering affine varieties $V \subseteq \C^m$
and $W\subseteq \C^n$.  We can see that $V\times W$ is an
affine variety in several ways.  The most direct way is
to proceed as follows.  Let $V\subseteq \C^m = \Spec(\C[x_1,\ldots,x_m])$ 
and $W \subseteq \C^n = \Spec(\C[y_1,\ldots,y_n])$.  Take
$\I(V) = \langle f_1, \ldots, f_s\rangle$  and
$\I(W) = \langle g_1, \ldots, g_t\rangle$.
Then since the $f_i$ and $g_j$ depend on separate sets of variables,
$V\times W =\V(f_1,\ldots,f_s,g_1,\ldots,g_t)$. This gives
a concrete set of equations defining the product 
$V\times W \subseteq \C^{m+n}$.

A fancier method is to use the mapping properties the product must
have.  This way also identifies the coordinate ring of the 
product variety in a more invariant form.
Namely, given affine varieties $V,W$, the product $V\times W$ (if it
exists) should be a variety with projections $\pi_1 : V\times W \to V$
and $\pi_2 : V \times W \to W$ such that whenever we have morphisms
$\phi : Z \to V$ and $\psi: Z \to W$ from some variety $Z$, 
there should be a unique morphism $\nu : Z \to V \times W$ such
that $\pi_1 \circ \nu = \phi$ and $\pi_2 \circ \nu = \psi$.  
On the level of coordinate rings, $\C[V \times W]$ should
be a $\C$-algebra with homomorphisms $i_1: \C[V] \to \C[V\times W]$
and $i_2: \C[W] \to \C[V\times W]$ (corresponding to the $\pi_i$).
Moreover, it should satisfy the property that whenever we have 
homomorphisms $j: \C[V] \to \C[Z]$ (corresponding to $\phi$)
and $k: \C[W]\to \C[Z]$ (corresponding to $\psi$), there should
exist a unique homomorphism $\ell: \C[V\times W] \to \C[Z]$
such that $j = \ell\circ i_1$ and $k = \ell\circ i_2$.
By the universal mapping property of the \emph{tensor
product} of $\C$-algebras, there exists exactly one algebra
that does this (up to isomorphism), 
namely $\C[V] \otimes_\C \C[W]$.  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} of Chapter~\ref{affine}. 
We could even take this as the definition of $V\times W$.
 
Once we have this, it is relatively routine to see that if $V$ 
is obtained by gluing together affine varieties $U_\a$ and $W$
is obtained by gluing together affines $U'_\b$, then
$V \times W$ is obtained by gluing together the 
$U_\a \times U'_\b$ in the corresponding fashion.
The details are presented in \cite{eh} and \cite{hartshorne}.

For instance, in Example~\ref{3.blowuppt}, we considered $\C^2 \times \P^1$.
This is obtained as a variety by gluing $V_0 = \Spec(\C[x,y]\otimes \C[u]) \simeq \C^3$
and $V_1 = \Spec(\C[x,y]\otimes\C[v]) \simeq \C^3$ together along the subsets
$V_{01} = \Spec(\C[x,y]\otimes\C[u]_u)$ and 
$\V_{10} = \Spec(\C[x,y]\otimes\C[v]_v)$
using the isomorphism $g_{01} : V_{01} \to V_{10}$ 
corresponding to the $\C$-algebra homomorphism 
defined by $v \mapsto 1/u$ and $x\mapsto x$, $y\mapsto y$.  

\subsection*{\emph{Separated Varieties}}
{}From the point of view of the classical topology,
if we allow all possible gluings, then we can obtain varieties
with some strange properties.  

\begin{example}
\label{3.nonsep}
For instance, as in Example~\ref{3.projline},
let $V_0 = \Spec(\C[u]) \simeq \C$, $V_1 = \Spec(\C[v]) \simeq \C$,
$V_{01} = \Spec(\C[u]_u) \simeq \Cstar$ and $V_{10} = \Spec(\C[v]_v) \simeq \Cstar$.
But now we take the gluing map $g_{01} : V_{01} \to V_{10}$ corresponding to the map 
of $\C$-algebras defined by $v \mapsto u$.  As before, as a set, the glued
variety $X$ consists of $\Cstar$ together with two additional points. 
However here we have a morphism $\pi: X \to \C$ whose fiber $\pi^{-1}(\{a\})$ over 
$a \in \Cstar$ contains one point, but whose fiber over $0$ consists of two 
points, $p_1$ corresponding to $0\in V_0$ and $p_2$ corresponding to $0\in V_1$.  
If $U_1,U_2$ are \emph{classical} open sets in $X$ with $p_1 \in U_1$ and 
$p_2\in U_2$, then $U_1\cap U_2\neq \emptyset$.  So the \emph{classical} 
topology on $X$ does not satisfy the Hausdorff property.
\eoe
\end{example}

Of course, it is easy to see that the \emph{Zariski} topology on a variety
is not Hausdorff except in one very special case (Exercise~\ref{3.ZariskiHausdorff}), but that 
is to be expected since Zariski open sets are so ``big''.

A different way to understand Example~\ref{3.nonsep} 
is to consider the product $X\times X$ and the \emph{diagonal mapping}
$\Delta : X \to X \times X$, defined on the set level 
by $\Delta(p) = (p,p)$ for all $p\in X$.
For $X$ from Example~\ref{3.nonsep}, it is not
difficult to see that $X\times X$ has a morphism
$\pi : X \times X \to \C^2$ which is one-to-one except over
$(0,0)$, and the fiber 
$\pi^{-1}((0,0)) = \{(p_1,p_1),(p_1,p_2),(p_2,p_1),(p_2,p_2)\}$.
Any Zariski closed subset of $X \times X$ that contains
one of these four points must contain all of them.  The image
of the diagonal mapping contains $(p_1,p_1)$ and $(p_2,p_2)$,
but not the other two, so it is not Zariski closed.  This
example motivates the following definition.

\begin{definition}
\label{3.sepdef}
We say a variety $X$ is \emph{\bfseries separated} if the 
image of the diagonal map $\Delta : X \to X\times X$ is
Zariski closed in $X\times X$.
\end{definition}

For instance, $\C^n$ is separated because the image of the 
diagonal in
$\C^{n} \times \C^n = \Spec(\C[x_1,\ldots,x_n,y_1,\ldots,y_n])$
is the affine variety $\V(x_1 - y_1, \ldots, x_n - y_n)$.

The connection between failure of separatedness and 
failure of the Hausdorff property in the classical topology seen 
in Example~\ref{3.nonsep} is a general phenomenon.

\begin{theorem}  A variety is separated if and only if
it is Hausdorff in the classical topology.  
\end{theorem}

Some additional important properties of separated varieties are
given in the following.

\begin{proposition}
\label{3.sepprops}
Let $X$ be a separated variety.
\begin{coxenum}
\item   If $f,g : Y \to X$ are morphisms, 
then $Z = \{y \in Y\ \vert\ f(y) = g(y)\}$ is a subvariety of $Y$.
\item If $U,V$ are affine open subsets of $X$, then $U\cap V$
is also affine.
\end{coxenum}
\end{proposition}

You will prove these statements in Exercise~\ref{3.seppropsex}.

Finally, we note that the requirement that $X$ is separated is often 
included in the definition of an abstract variety.  When that
is done, what we have called a variety is sometimes called a
\emph{pre-variety}. 


\exercises

\small
\smallskip

\begin{exercise}
\label{3.manifold}   
\begin{thmenum}
\item Let $X$ be a $C^m$ differentiable manifold.  
If $U \subseteq X$, what does it mean for $f : U \to \R$ to 
be a $C^m$ differentiable function?  Show that assigning
$\cO_X(U)$ to be the $\R$-algebra of all $\C^m$ differentiable
functions defines a sheaf $\cO_X$ and makes 
$(X,\cO_X)$ into a ringed space over $\R$.   
\item Let $(X,\cO_X)$ be a ringed space over $\R$.
Show that if every point in $X$ has a neighborhood $U$
such that $(U,\cO_X|_U)$ is isomorphic as a ringed space
over $\R$ to $(V,\cO_V^m)$ for some $V \subseteq \R^n$, then
$X$ has the structure of a $C^m$ differentiable $n$-manifold.
\end{thmenum}
\end{exercise}

\begin{exercise}  
Let $V = \Spec(R)$ be an affine variety.
Show that a collection of open sets $V_{f_\a}$, $\a \in A$,
covers $V$ if and only if the ideal 
$\langle f_\a\ |\ \a\in A\rangle$ is the whole ring $R$.
Hint: Think about the proof of part a of Proposition~\ref{3.strucsheafprops}.
\end{exercise}

\begin{exercise}
\label{3.quasicompact}
Let $V = \Spec(R)$ be an affine variety. 
\begin{thmenum}
\item Show that every ideal $I \subseteq R$ can be written in
the form $I = \langle f_1,\ldots,f_s\rangle$, where $f_i \in R$.
(This is the Hilbert Basis Theorem in $R$.)
\item Let $W \subseteq V$ be a subvariety.  Show that the complement
of $W$ in $V$ can be written as a union of a finite collection
of open affine sets of the form $V_f$.  
\item Deduce that every open cover of $V$ (in the Zariski topology)
has a finite subcover.  (In topological language, this says that affine varieties 
are \emph{quasicompact} in the Zariski topology.)
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{3.prinaffineopens}
In this exercise, you will verify that the sheaf properties
from Definition~\ref{3.sheafdef} hold for the structure
sheaf of an affine variety $V = \Spec(R)$, if we restrict our attention
to open sets of the form $V_f$ for $f \in R$.  
\begin{thmenum}
\item Let $U = V_f$ and $U_\a = V_{f_\a}$ for some 
$f_\a \in R$. Show that if $u \in \cO_V(U) = R_f$ restricts
to $0$ in all $\cO_V(U_\a) = R_{f_\a}$, then $u = 0$.  
\item Show that if we have $u_\a \in \cO_V(U_\a) = R_{f_\a}$
such that for all $\a$ and $\b$, the restrictions of
$u_\a$ and $u_\b$ are equal in 
$\cO_V(U_\a \cap U_\b) = R_{f_\a f_\b}$,
then there is some $u \in \cO_V(U)$ that restricts to $u_\a$ for 
all $\a$.
\end{thmenum}
\end{exercise}


\begin{exercise}
\label{3.subvariety}
(Subvarieties of abstract varieties.)  Let $X$ be an abstract
variety, and let $Y \subseteq X$ be a Zariski closed subset. 
If $U\subseteq Y$
is open, let $\cO_Y(U)$ to be the set of all $\phi : U \to \C$
such that for all $p\in U$, there exist $U_1$ open in $X$ and
and a regular function $\phi_1 : U_1 \to \C$ in $\cO_X(U_1)$ with 
$p \in U_1\cap Y \subseteq U$ and $\phi|_{U_1 \cap Y} = \phi_1|_{U_1 \cap Y}$.
\begin{thmenum}
\item Show that $\cO_Y$ is a sheaf of $\C$-algebras on $Y$.
\item When $X$ is an affine variety and $Y$ is a closed subset,
show that $\cO_Y$ coincides with the structure sheaf of $Y$.
\item Show that if $(X,\cO_X)$ is an abstract variety, then
$(Y,\cO_Y)$ also has the structure of an abstract variety.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{3.globalsecs}
Show that the only regular functions on all of $\P^n$
are the constant functions, that is, $\cO_{\P^n}(\P^n) = \C$.
\end{exercise}

\begin{exercise}
\label{3.funcfieldprops}
Let $X$ be an irreducible abstract variety and $U \subset X$ a
Zariski open subset.
\begin{thmenum}
\item Let $f,g$ be rational functions on $X$.
Show that $f \sim g$ if $f|_U = g|_U$ is
an equivalence relation.
\item Show that the set of equivalence classes of the relation
in part a has the structure of a field.
\item Show that if $U \subseteq X$ is an affine open subset
of $X$, then $\C(U) \simeq \C(X)$.  
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{3.projplane}
Let $V_0 = \Spec(\C[x,y])$, $V_1 = \Spec(\C[z,w])$
and $V_2 = \Spec(\C[r,s])$.  Let $V_{01} = \Spec(\C[x,y]_x)$,
$V_{10} = \Spec(\C[z,w]_z)$, 
$V_{02} = \Spec(\C[x,y]_y)$, $V_{20} = \Spec(\C[r,s]_r)$,
$V_{12} = \Spec(\C[z,w]_w)$, and $V_{21} = \Spec(\C[r,s]_s)$.
Let $g_{01}: V_{01} \to V_{10}$ be morphism corresponding to
the $\C$-algebra homomorphism defined by $z\mapsto 1/x$ and $w\mapsto y/x$.  
Let $g_{02}: V_{02} \to V_{20}$ be defined by 
$r\mapsto 1/y$ and $s\mapsto x/y$.  Finally let
$g_{12} : V_{12} \to V_{21}$ be defined by
$r\mapsto z/w$ and $s\mapsto 1/w$.  
\begin{thmenum}
\item Show that each $g_{ij}$ defined above is an isomorphism, 
and let $g_{ji} = g_{ij}^{-1}$.
\item Show that the whole collection of mappings $\{g_{ij}\}$ satisfies
the compatibility conditions for gluing.
\item Identify the variety $X$ obtained by gluing the $\{V_i\}$
along the $\{V_{ij}\}$ using the $\{g_{ij}\}$.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{3.ZariskiHausdorff}
Show that the Zariski topology on a variety $X$
is Hausdorff if and only if $X$ is \emph{finite} as a set.
\end{exercise}

\begin{exercise} 
Let $V = \Spec(R)$ be an affine variety.  Which 
$\C$-algebra homomorphism $R\otimes_\C R \to R$
corresponds to the diagonal mapping
$\Delta : V \to V \times V$?  Prove your assertion.  Hint: 
consider the universal mapping property of $V\times V$.
\end{exercise}

\begin{exercise}
\label{3.seppropsex}
\begin{thmenum}
\item Prove part a of Proposition~\ref{3.sepprops}.
Hint: Show first that if $F : Y \to X\times X$
is defined by $F(y) = (f(y),g(y))$, then $Z = F^{-1}(\Delta(X))$. 
\item Prove part b of Proposition~\ref{3.sepprops}.  Hint:  
Show first that $U \cap V$ is identified with $\Delta(X) \cap U\times V
\subseteq X\times X$.
\end{thmenum}
\end{exercise}

\normalsize
\section{Fans and Normal Toric Varieties}

In this section we present the construction of the 
toric variety $X_\Sigma$ corresponding to a fan $\Sigma$.
We will also relate the varieties obtained to many examples we 
have encountered previously, and we will show how properties of
the fan relate to properties such as smoothness and compactness of 
$X_\Sigma$.  A toric variety in this context means the same thing as in
Chapters 1 and 2, although we now allow abstract varieties as in
\S 0.  

\begin{definition}
\label{3.abstoric}
A \emph{toric variety} is an irreducible variety $X$ 
containing a torus $T_N \simeq (\Cstar)^n$ as a
Zariski open subset such that the action of $T_N$
on itself extends to an action of $T_N$ on $X$.  
\end{definition}

Let $\Sigma$ be a fan in $N_\R$, a collection of 
strongly convex rational polyhedral cones with the property
that each face of $\sigma$ in $\Sigma$ is also
in $\Sigma$, and the intersection $\sigma_1\cap\sigma_2$
of any two cones in $\Sigma$ is a face of each.
Then from Chapter~\ref{affine}, Theorem~\ref{1.coneafftor},
for each cone $\sigma$ in $\Sigma$, we have the affine toric variety 
\[
V_\sigma = \Spec(\C[\sS_\sigma]) = \Spec(\C[\sigma^\vee \cap M]).
\]
Moreover, if $\tau$ is a face of $\sigma$, then by Chapter~\ref{affine},
Definition~\ref{1.facedef}, there is some $m \in \sigma^\vee$
such that $\tau = \sigma \cap H_m$ where 
$H_m = \{u\in N_\R \mid \langle m,u\rangle = 0\}$ is the 
hyperplane normal to $m$.  

The following proposition gives several additional properties
of the $\sS_\sigma$ and their semigroup rings.  

\begin{proposition}
\label{3.semiprops}
Let $\sigma, \sigma_1, \sigma_2$ be cones in a fan $\Sigma$.  
\begin{thmenum}  
\item If $m \in M$ and $\tau = \sigma \cap H_m$ is the corresponding
face of $\sigma$, then 
\[
\sS_\tau = \sS_\sigma + \Z_{\ge 0}\cdot (-m).
\]
\item In the situation of part a, $\C[\sS_\tau]$ equals the
localization $\C[\sS_\sigma]_{\chi^m}$.
\item If $\sigma_1\cap \sigma_2 = \tau$, then there exists 
$m \in \sigma_1^\vee \cap (-\sigma_2)^\vee$
such that $\sigma_1\cap H_m = \tau = \sigma_2\cap H_m$.
\item In the situation of part c, $\sS_\tau = \sS_{\sigma_1} + \sS_{\sigma_2}$.
\end{thmenum}
\end{proposition}

Before giving the proof, we indicate some of the intuition behind these
statements with a simple example.  

\begin{example}
\label{3.semipropsex}
Let $\sigma_1 = \Cone(e_1+e_2,e_2)$ (as in Exercise~2.8 of Chapter~\ref{affine}),
and let $\sigma_2 = \Cone(e_1,e_1+e_2)$ in $N_\R = \R^2$.  Then 
$\tau = \sigma_1 \cap \sigma_2 = \Cone(e_1+e_2)$.  We show
the dual cones $\sigma_1^\vee = \Cone(e_1,-e_1+e_2)$, 
$\sigma_2^\vee = \Cone(e_1-e_2,e_2)$, 
and $\tau^\vee = \sigma_1^\vee + \sigma_2^\vee$
in the following figure.  The darkest shaded region on the right
is $\sigma_1^\vee \cap \sigma_2^\vee$, and $\tau^\vee = \sigma_1^\vee + \sigma_2^\vee$
(which coincides with $\sigma_1^\vee \cup \sigma_2^\vee$ in this case).  
\begin{figure}[ht]
\begin{center}
\includegraphics[height=2.0in]{ch3fig1.ps}
\end{center}
\caption{The cones $\sigma_1,\sigma_2,\tau$ and their duals} 
\end{figure}

We have $\tau = \sigma_1 \cap H_m = \sigma_2 \cap H_{-m}$ for 
$m = -e_1 + e_2 \in \sigma_1^\vee$
and $-m = e_1-e_2 \in \sigma_2^\vee$.  Moreover, 
$\tau^\vee = \sigma_1^\vee + \sigma_2^\vee$,
so as in part a of the proposition, using $\sigma_1$,
\[
\sS_\tau = \sS_{\sigma_1} + \Z_{\ge 0}\cdot (-m).
\]  
Then as in part b,
\[
\C[\sS_\tau] = \C[x,x^{-1}y,xy^{-1}] = \C[\sS_{\sigma_1}]_{xy^{-1}}.
\]
There is also clearly a corresponding statement for $\sigma_2$. 
Part c holds if we take $m = -e_1 + e_2$, and we can see part d
clearly from the figure because $\sS_\tau$ will contain
all the lattice points in $\sigma_1^\vee \cap M$ and $\sigma_2^\vee\cap M$.
\eoe
\end{example}
 
\begin{proof}
For part a, note that the inclusion $\tau \subseteq \sigma$ implies
that $\sigma^\vee \subseteq \tau^\vee$, so $\sS_\sigma \subseteq \sS_\tau$.
Since $\tau = \sigma \cap H_m$, $\Z \cdot (-m) \subseteq \sS_\tau$ as
well, and we have $\sS_\sigma + \Z_{\ge 0}\cdot (-m) \subseteq \sS_\tau$.
To see the opposite inclusion, if $p \in \sS_\tau$, then in Exercise~\ref{3.conelemma}
you will show that for some $\ell \in \Z_{\ge 0}$,  $p + \ell m$ lies in $\sigma^\vee\cap M$.
Hence $p \in \sS_\sigma + \Z_{\ge 0}\cdot (-m)$.

Part b follows immediately from part a.

For part c, let $\lambda = \sigma_1 + (-\sigma_2)$, which is another rational
polyhedral cone in $N_\R$.  If $m$ is any vector in the relative interior
of $\lambda^\vee$, then $\lambda \cap H_m$ is the minimal face 
of the cone $\lambda$, namely $\lambda \cap -\lambda$ 
(Proposition~\ref{1.dualfaceprops} of Chapter~\ref{affine}).  We claim any such $m$ satisfies 
the desired conditions.  First, $\sigma_1 \subset \lambda$, so 
$m \in \sigma_1^\vee$.  Similarly, $m \in (-\sigma_2)^\vee$.  Next, 
$\tau$ is contained in $\lambda \cap (-\lambda)$, so $\tau$ is contained
in $\sigma_1 \cap H_m$.  On the other hand, if $x \in \sigma_1 \cap H_m$, 
then $x = w - v$ for $v \in \sigma_1$ and $w \in \sigma_2$.  Then
$x + v = w$ is in $\sigma_1\cap \sigma_2 = \tau$.  But if $x + v\in\tau$,
for $x,v\in \sigma_1$ then $x$ and $v$ must also be in $\tau$.  Hence
$\sigma_1 \cap H_m = \tau$, and the same argument applied to $\sigma_2$
shows $\sigma_2 \cap H_m = \tau$ as well.

Finally, we must show part d.  The inclusion $\sS_{\sigma_1} + \sS_{\sigma_2} \subseteq \sS_\tau$
follows directly from the property 
$\sigma_1^\vee + \sigma_2^\vee  = (\sigma_1 \cap \sigma_2)^\vee = \tau^\vee$.
The reverse inclusion then follows from parts a and c.  First, 
the $m$ in part c can be taken in $M$ since all the cones in question are
rational.   If $p \in \sS_\tau$, then
$p = q + \ell (-m)$ for some $\ell\in \Z_{\ge 0}$ and $-m \in (-\sigma_2)^\vee$
implies $m \in \sigma_2^\vee \cap M = \sS_{\sigma_2}$.  So 
$p \in \sS_{\sigma_1} + \sS_{\sigma_2}$.
\end{proof}

Now if $\sigma_1, \sigma_2$ are cones in the fan $\Sigma$ with $\sigma_1\cap \sigma_2 = \tau$, 
then by part b of Proposition~\ref{3.semiprops}, the injective $\C$-algebra homomorphisms
\[
\C[\sS_{\sigma_1}] \hookrightarrow \C[\sS_\tau] \hookleftarrow \C[\sS_{\sigma_2}]
\]
yield affine opens 
\[
V_{\sigma_1, \sigma_2} = \Spec(\C[V_{\sigma_1}]_{\chi^m}) \subseteq V_{\sigma_1}
\]
and 
\[
V_{\sigma_2, \sigma_1} = \Spec(\C[V_{\sigma_2}]_{\chi^{-m}}) \subseteq V_{\sigma_2}.
\]
Since each of these is isomorphic to the affine variety $V_\tau$, we have
isomorphisms $g_{\sigma_1,\sigma_2} : V_{\sigma_1,\sigma_2} \simeq V_{\sigma_2,\sigma_1}$
for all pairs of cones $\sigma_1,\sigma_2$.  In Exercise~\ref{3.fangluing},
you will show that the compatibility conditions for gluing the affine 
varieties $V_\sigma$ along the subvarieties $V_{\sigma_1,\sigma_2}$ are satisfied.
Hence, we obtain an abstract variety $X_\Sigma$ associated to the fan $\Sigma$.

\begin{theorem}
\label{3.toricfromfan}
The variety $X_\Sigma$ constructed as above is a normal, separated 
toric variety.
\end{theorem}

\begin{proof}
Since each cone in a fan is strongly convex, $\tau = \{0\}\subseteq N$ is a face of all
$\sigma \in \Sigma$.  Hence we have $T(N) = \Spec(\C[M]) \simeq (\Cstar)^n \subseteq V_\sigma$
for all $\sigma$.  These tori are all identified by the gluing, so we have
$T(N) \subseteq X_\Sigma$.  It is not difficult to check that the gluing isomorphisms
$g_{\sigma_1,\sigma_2}$ are $T(N)$-equivariant, so the action of $T(N)$ on itself
extends to an action on $X_\Sigma$.
 
By Theorem~{1.satthm} of Chapter~\ref{affine}, for each $\sigma \subseteq \Sigma$,
$V_\sigma$ is a normal affine variety.  Hence the variety $X_\Sigma$ is normal by 
Proposition~\ref{3.normalcover}.

To see that $X_\Sigma$ is separated it will suffice to show that for each 
pair of cones $\sigma_1,\sigma_2$ in $\Sigma$, 
writing $\tau = \sigma_1 \cap\sigma_2$, the image of the diagonal map 
$\Delta : V_{\tau} \to V_{\sigma_1} \times V_{\sigma_2}$ is Zariski closed 
(Exercise~\ref{3.sepcover}). We note first that by part d of 
Proposition~\ref{3.semiprops}, $V_\tau = \C[\sS_\tau] = \C[\sS_{\sigma_1} + \sS_{\sigma_2}]$.
Then $\Delta$ is the morphism corresponding to the $\C$-algebra homomorphism 
\[
\delta : \C[\sS_{\sigma_1}] \otimes_\C \C[\sS_{\sigma_2}] \to \C[\sS_{\sigma_1} + \sS_{\sigma_2}]
\]
defined by $\delta(\chi^{m_1} \otimes \chi^{m_2}) = \chi^{m_1 + m_2}$ (and extended by 
linearity to general elements of the tensor product).  It follows that 
the image of $\Delta$ is a Zariski closed subset of $V_{\sigma_1} \times V_{\sigma_2}$.
\end{proof}



\exercises
\small
\smallskip

\begin{exercise}
\label{3.conelemma}
Show that if $\tau = \sigma \cap H_m$ is a face of a strongly 
convex rational polyhedral 
cone $\sigma$, then for all $p \in \tau^\vee$, there
exist $\ell \in \Z_{\ge 0}$ such that $p + \ell m \in \sigma^\vee$.
\end{exercise}  

\begin{exercise}
\label{3.fangluing}
Let $\Sigma$ be a fan in $N_\R$.
Show that the isomorphisms $g_{\sigma_1,\sigma_2}$ satisfy the 
compatibility conditions for gluing the $V_\sigma$ along
the subvarieties $V_{\sigma_1,\sigma_2}$ from \S 0. 
\end{exercise}

\begin{exercise}
\label{3.sepcover} 
Let $X$ be a variety obtained by gluing the affine open subsets
$\{V_\a\}$ along open subsets $V_{\a\b} \subseteq V_\a$ 
by isomorphisms $g_{\a\b} : V_{\a\b}\simeq V_{\b\a}$.
Show that if the image of $\Delta : V_{\a\b} \to V_\a \times V_\b$
defined by $\Delta(p) = (p,g_{\a\b}(p))$ is Zariski closed for all $\a,\b$,
then $X$ is separated.
\end{exercise}


\begin{itemize}
    \item{Sumihiro theorem -- say separated}
    \item{fan gives separated normal toric variety}
    \item{review previous examples, including Hirzebruch surface}
    \item{weighted proj space via $N = \Z^{n+1}/(\Z(q_0,\dots,q_n))$}
    \item{$\xs$ smooth $\leftrightarrow \Sigma$ smooth}
    \item{$\xs$ orbifold $\leftrightarrow \Sigma$ simplicial}
    \item{$\xs$ compact in classical top $\leftrightarrow \Sigma$ complete}
\end{itemize}

\section{The Orbit--Cone Correspondence}
 
\begin{itemize}
    \item{$\P^2$}
    \item{limits of one parameter subgroups}
    \item{recovering fan from one parameter subgroups}
    \item{$V(xy-zw)$}
\end{itemize}

\section{Equivariant Maps of Toric Varieties}
 
\begin{itemize}
    \item{maps $\leftrightarrow \Sigma \rightarrow \Sigma'$}
    \item{change of lattice}
    \item{star and blowups, relate to truncated pyramid}
    \item{fibration s.e.s}
\end{itemize}

\section{Complete, Proper, and Separated}

\begin{itemize}
    \item{complete and proper}
    \item{separated, separation lemma}
\end{itemize}
