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

\setcounter{section}{-1}

\section{Background:\ Abstract Varieties}

The projective toric varieties studied in Chapter~\ref{poly} are
unions of Zariski open sets, each of which is an affine variety.  We
will begin this chapter by building up to a general construction of
abstract varieties obtained by gluing together affine varieties in an
analogous way.  The resulting varieties will be \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.

Since we make extensive use of localization, we will work exclusively
with irreducible affine varieties.  Sheaf theory, while important for
later chapters, will make only a modest appearance here.  For a more
general approach to the concept of abstract variety, we recommend
standard books such as \cite{eh}, \cite{hartshorne} or
\cite{shafarevich}.

\vspace*{-1pt}

\subsection*{\emph{Regular Functions}}
Let $V = \Spec(R)$ be an irreducible affine variety.  In \S0 of
Chapter 1, we defined the Zariski open subset $V_f = V\ \setminus\
\V(f) \subseteq V$ for $f \in R$ and showed that $V_f = \Spec(R_f)$,
where $R_f$ is the localization of $R$ at $f$.  The open sets $V_f$
form a \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}).

For an affine variety, a morphism $V \to \C$ is called a \emph{regular
map}, so that the coordinate ring of $V$ consists of all regular maps
from $V$ to $\C$.  We now define what it means to be regular on an
open subset of $V$.

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

\bigbreak\goodbreak

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

Here are some cases where $\cO_V(U)$ is easy to compute.

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

\begin{proof}
It is clear from Definition~\ref{3.regfundef} that elements of $R$
define regular functions on~$V$, hence elements of $\cO_V(V)$.
Conversely, if $\phi \in \cO_V(V)$, then for all $p \in V$ there is
$f_p \in R$ such that $p \in V_{f_p}$ and $\phi = a_p/f_p^{\,n_p}\in
R_{f_p}$.  The ideal $I = \langle f_p^{\,n_p} \mid p \in V\rangle
\subseteq R$ 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 $\phi = \sum_{p\in S} g_p f_p^{\,n_p}\phi = \sum_{p\in S} g_p\,
a_p \in R$, as desired.

For part (b), let $U \subseteq V_f$ be Zariski open.  Then $U$ is
Zariski open in $V$, and whenever $g \in R$ satisfies $V_g \subseteq
U$, we have $V_g = V_{fg}$ with coordinate ring
\[
R_{fg} = (R_f)_{g/f^\ell}
\]
for all $\ell \ge 0$.  These observations easily imply that 
\begin{equation}
\label{3.restrictvf}
\cO_V(U) =\cO_{V_f}(U).
\end{equation}
Then setting $U = V_f$ gives
\[
\cO_V(V_f) = \cO_{V_f}(V_f) = R_f,
\]
where the last equality follows by applying part (a) to $V_f =
\Spec(R_f)$.
\end{proof}

\subsection*{\emph{Local Rings}}
When $V = \Spec(R)$ is an irreducible affine variety, we can describe
regular functions using the \emph{local rings} $\cO_{V,p}$ introduced
in \S0 of Chapter~\ref{affine}.  A rational function in $\C(V)$ is
contained in the local ring $\cO_{V,p}$ precisely when it is regular
in a neighborhood of $p$.  It follows that whenever $U \subseteq V$ is
open, we have
\[
\bigcap_{p\in U} \cO_{V,p} = \cO_V(U).
\]
Thus regular functions on $U$ are rational functions on $V$ that are
defined everywhere on $U$.  In particular, when $U = V$,
Proposition~\ref{3.strucsheafprops} implies that
\begin{equation}
\label{3.intlocalrings}
\bigcap_{p\in V} \cO_{V,p} = \cO_V(V) = R = \C[V].
\end{equation}

\subsection*{\emph{The Structure Sheaf}} Given an irreducible affine
variety $V$, the operation 
\[
U \mapsto \cO_V(U),\qquad U \subseteq V \text{ open},
\]
has the following useful properties:
\begin{coxitem} 
\item When $W \subseteq U$, Definition~\ref{3.regfundef} shows that
there is an obvious restriction map
\[
\rho_{U,W} : \cO_V(U) \to \cO_V(W)
\]
defined by $\rho_{U,W}(\phi) = \phi\res{W}$.  It follows that
$\rho_{U,U}$ is the identity map and that $\rho_{V,W}\circ\rho_{U,V}
= \rho_{U,W}$ whenever $W \subseteq V \subseteq U$.
\item If $\{U_\alpha\}$ is an open cover of $U \subseteq V$, then the
sequence 
\[
0 \longrightarrow \cO_V(U) \longrightarrow \prod_{\a}
\cO_V(U_\a)\
\lower3pt\hbox{$\stackrel{\displaystyle\longrightarrow}{\longrightarrow}$}
\prod_{\a,\b} \cO_V(U_\a\cap U_\b)
\]
is exact.  Here, the second arrow is defined by the restrictions
$\rho_{U,U_\a}$ and the double arrow is defined by
$\rho_{U_\a,U_\a\cap U_\b}$ and $\rho_{U_\b,U_\a\cap U_\b}$.
Exactness at $\cO_V(U)$ means that regular functions are determined
locally (that is, two regular functions on $U$ are equal if their
restrictions to all $U_\a$ are equal), and exactness at $\prod_{\a}
\cO_V(U_\a)$ means that regular functions on the $U_\a$ agreeing on
the overlaps $U_\a \cap U_\b$ patch together to give a regular
function on $U$.
\end{coxitem}
In the language of sheaf theory, these properties imply that $\cO_V$
is a \emph{sheaf} of $\C$-algebras, called the \emph{structure sheaf}
of $V$.  We call $(V,\cO_V)$ a \emph{ringed space over}~$\C$.  Also,
since \eqref{3.restrictvf} holds for all open sets $U \subseteq V_f$, we
write
\[
\cO_V\res{V_f} = \cO_{V_f}.
\]
In terms of ringed spaces, this means $(V_f,\cO_V\res{V_f}) =
(V_f,\cO_{V_f})$. 

\subsection*{\emph{Morphisms}} By \S 0 of Chapter~\ref{affine}, a
polynomial mapping $F : V_1 \to V_2$ between affine varieties
corresponds to the $\C$-algebra homomorphism $F^* : \C[V_2] \to
\C[V_1]$ defined by $F^*(g) = g \circ F$ for $g \in \C[V_2]$.  We now
extend this to open sets of affine varieties.

\begin{definition}
\label{3.mordef1}
Let $U_i \subseteq V_i$ be Zariski open subsets of irreducible affine
varieties for $i = 1,2$.  A function $\Phi : U_1 \to U_2$ is a
\emph{\bfseries morphism} if $\phi \mapsto \phi\circ\Phi$ defines a
map
\[
\Phi^* : \cO_{V_2}(U_2) \longrightarrow \cO_{V_1}(U_1).
\]
\end{definition}

Thus $\Phi : U_1 \to U_2$ is a morphism if composing $\Phi$
with regular functions on $U_2$ gives regular functions on $U_1$.  Note
also that $\Phi^*$ is a $\C$-algebra homomorphism since it comes from
composition of functions.

\begin{example}
\label{3.morphismex}
Suppose that $\Phi: V_1 \to V_2$ is a morphism according to
Definition~\ref{3.mordef1}.  If $V_i = \Spec(R_i)$, then the above map
$\Phi^*$ gives the $\C$-algebra homomorphism
\[
R_2 = \cO_{V_2}(V_2) \longrightarrow \cO_{V_1}(V_1) = R_1.
\]
By Chapter~\ref{affine}, the homomorphism $R_2 \to R_1$ gives a map of
affine varieties $V_2 \to V_1$.  In Exercise~\ref{3.morphismexer} you
will show that this is the map $\Phi : V_1 \to V_2$ we started with.\eoe
\end{example}

Example~\ref{3.morphismex} shows that when we apply
Definition~\ref{3.mordef1} to maps between affine varieties, we get
the same morphisms as in Chapter~\ref{affine}.  In
Exercise~\ref{3.morphismexer} you will verify the following properties
of morphisms:
\begin{coxitem}\label{3.morphismprops}
\item If $U$ is open in an affine variety $V$, then
\[
\cO_V(U) = \{ \phi : U \to \C \mid \phi \text{ is a morphism}\}.
\]
Hence regular functions on $U$ are just morphisms from $U$ to $\C$.
\item A composition of morphisms is morphism.
\item An inclusion of open sets $W \subseteq U$ of an affine variety
  $V$ is a morphism.
\end{coxitem}

These observations enable us to understand what it means for $\Phi :
U_1 \to U_2$ be a morphism when $U_i \subseteq V_i$ are arbitrary open
sets.  Let $V_i = \Spec(R_i)$ and write $U_1 = \bigcup_{f \in S}
(V_1)_f$ for $S \subseteq R_1$ finite.  Then $(V_1)_{f} \subseteq
U_1$ and $U_2 \subseteq V_2$ gives the morphism
\[
(V_1)_{f} \subseteq U_1 \stackrel{\Phi}{\longrightarrow} U_2
\subseteq V_2.
\]
This is given by a $\C$-algebra homomorphism $R_2 \to (R_1)_{f}$.
Since the $(V_1)_f$ cover $U_1$, we see that locally, morphisms look
like the polynomial maps introduced in Chapter~\ref{affine}.  In
particular, morphisms are continuous in the Zariski topology.

We say that a morphism $\Phi: U_1 \to U_2$ is an \emph{isomorphism} if
$\Phi$ is bijective and its inverse function $\Phi^{-1}: U_2 \to U_1$
is also a morphism.

\subsection*{\emph{Gluing Together Affine Varieties}} We now are ready
to define abstract varieties by gluing together open subsets of affine
varieties.  The model is what happens for $\P^n$.  Recall from \S0 of
Chapter~\ref{poly} that $\P^n$ is covered by open sets 
\[
U_i =  \P^n \setminus \V(x_i) =
\Spec\big(\C\big[{\textstyle \frac{x_1}{x_i},\ldots,\frac{x_{i-1}}{x_i},
\frac{x_{i+1}}{x_i},\ldots,\frac{x_n}{x_i}}\big]\big)
\]
for $i = 0,\dots,n$.  Each $U_i$ is a copy of $\C^n$ that uses a
different set of variables.  Given $i \ne j$, these copies of $\C^n$
are ``glued together'' as follows.  We have open subsets
\begin{equation}
\label{3.uiuj}
(U_i)_{\frac{x_j}{x_i}} \subseteq U_i \quad\text{and}\quad
(U_j)_{\frac{x_i}{x_j}} \subseteq U_j,
\end{equation}
and we also have the isomorphism
\begin{equation}
\label{3.gluepn}
g_{ij} : (U_i)_{\frac{x_j}{x_i}} \simeq (U_j)_{\frac{x_i}{x_j}}
\end{equation}
since both give the same open set $U_i\cap U_j$ in $\P^n$.  At the
level of coordinate rings, this comes from the isomorphism
\[
g_{ij}^* : \C\big[{\textstyle
\frac{x_1}{x_j},\ldots,\frac{x_{j-1}}{x_j},
\frac{x_{j+1}}{x_j},\ldots,\frac{x_n}{x_j}}\big]_{\frac{x_i}{x_j}}
\simeq \C\big[{\textstyle \frac{x_1}{x_i},\ldots,\frac{x_{i-1}}{x_i},
\frac{x_{i+1}}{x_i},\ldots,\frac{x_n}{x_i}}\big]_{\frac{x_j}{x_i}}
\]
defined by
\[
{\textstyle \frac{x_k}{x_j}} \longmapsto {\textstyle \frac{x_k}{x_i}}
\text{\large$/$}{\textstyle \frac{x_j}{x_i}}\quad (k \ne j)
\quad\text{and} \quad
\big({\textstyle \frac{x_i}{x_j}}\big)^{-1} \longmapsto {\textstyle
\frac{x_j}{x_i}}.
\]

We can turn this around and start from the affine varieties $U_i
\simeq \C^n$ given above and glue together the open sets in
\eqref{3.uiuj} using the isomorphisms $g_{ij}$ from \eqref{3.gluepn}.
This gluing is consistent since $g_{ji} = g_{ij}^{-1}$ and $g_{ik} =
g_{ij}\circ g_{jk}$ wherever all three maps are defined.  The result
of this gluing is the projective space $\P^n$.

To generalize this, suppose we have a finite collection $\{V_\a\}_\a$
of irreducible affine varieties and for all pairs $\a,\b$ 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 following compatibility
conditions:
\begin{coxitem}
\item  $g_{\b\a} = g_{\a\b}^{-1}$ for all pairs $\a, \b$.
\item $g_{\a\b}(V_{\a\b}\cap
V_{\a\g}) = V_{\b\a}\cap V_{\b\g}$ and $g_{\a\g} = g_{\b\g}\circ
g_{\a\b}$ on $V_{\a\b}\cap V_{\a\g}$ for all triples $\a,\b,\g$.
\end{coxitem}

We are now ready to glue.  Let $Y$ be the disjoint union of the $V_\a$
and define a relation $\sim$ on $Y$ by $a \sim b$ if and only if $a
\in V_\a$, $b \in V_\b$ for some $\a,\b$ with $b = g_{\a\b}(a)$.  The
first compatibility condition shows that $\sim$ is reflexive and
symmetric; the second shows that it is transitive.  Hence $\sim$ is an
equivalence relation and we can form the quotient space $X = Y/\sim$
with the quotient topology.  For each $\a$, let
\[
U_\a = \{ [a] \in X \mid a \in V_\a\}.
\]
Then $U_\a \subseteq X$ is an open set and the map $h_\a(a) = [a]$
defines a homeomorphism $h_\a : V_\a \simeq U_\a \subseteq X$.  Thus
$X$ locally looks like an irreducible affine variety.

\begin{definition}
\label{3.absvardef}
We call $X$ the \emph{\bfseries abstract variety} determined by the above data.
\end{definition}

An abstract variety $X$ comes equipped with the Zariski topology whose
open sets are those sets that restrict to open sets in each $U_\a$.
The Zariski closed subsets $Y \subseteq X$ are called
\emph{subvarieties} of $X$.  We say that $X$ is \emph{irreducible} is
it is not the union of two proper subvarieties.  Since $X$ is locally
irreducible, it is irreducible if and only if it is connected.  

\subsection*{\emph{The Structure Sheaf of an Abstract Variety}} Let
$U$ be an an open subset of an abstract variety $X$ and set $W_\a =
h_\a^{-1}(U\cap U_a) \subseteq V_\a$.  Then a function $\phi : U \to
\C$ is \emph{regular} if
\[
\phi\circ h_\a\res{W_\a} : W_\a \longrightarrow \C
\]
is regular for all $\a$.  The compatibility conditions ensure that this is
well-defined, so that one can define
\[
\cO_X(U) = \{\phi : U \to \C \mid \phi \text{ is regular}\}.
\]
This gives the \emph{structure sheaf} $\cO_X$ of $X$.  Thus an
abstract variety is really a ringed space $(X,\cO_X)$ with a finite
open covering $\{U_\a\}_\a$ such that $(U_\a,\cO_X\res{U_\a})$ is
isomorphic to the ringed space $(V_\a,\cO_{V_\a})$ of the affine
variety $V_\a$.  (We leave the definition of isomorphism of ringed
spaces to the reader.)

\begin{example}
\label{3.projective}
We saw above that $\P^n$ can be obtained by gluing together the open
sets \eqref{3.uiuj} using the isomorphisms $g_{ij}$ from
\eqref{3.gluepn}.  This shows that $\P^n$ is an abstract variety with
affine open subset $U_i \subseteq \P^n$.  More generally, given an
irreducible projective variety $V \subseteq \P^n$, we can cover $V$
with affine open subset $V\cap U_i$, and the gluing implicit in
equation \eqref{2.vuiuj} from Chapter~\ref{poly}.  We conclude that
irreducible projective varieties are also abstract varieties.

In a similar way, $\P^n \times \C^m$ can be viewed as gluing affine
spaces $U_i \times \C^m \simeq \C^{n+m}$ along suitable open subsets.
Thus $\P^n \times \C^m$ is an abstract variety, and the same is true
for irreducible subvarieties $V \subseteq \P^n \times \C^m$.  \eoe
\end{example}

In fact, \emph{all} of the various types of ``irreducible variety''
introduced so far in this book can be subsumed under the concept of
``irreducible abstract variety.''  Even more generally, one can define
abstract varieties that locally look like arbitrary affine varieties,
not necessarily irreducible (see \cite{eh,hartshorne,shafarevich}).

\begin{example}
\label{3.blowuppt}
Consider $V_0 = \C^2 = \Spec(\C[u,v])$ and $V_1 = \C^2 = \Spec(\C[w,z])$
with
\begin{align*}
V_{01} &= V_0 \setminus \V(v) = \Spec(\C[u,v]_v)\\
V_{10} &= V_1 \setminus \V(z) = \Spec(\C[w,z]_z)
\end{align*}
and gluing data 
\begin{align*}
g_{01} &: V_{01} \to V_{10}\ \text{ coming from 
the $\C$-algebra homomorphism }\\ g_{01}^* &:\C[w,z]_z
\to \C[u,v]_v\ \text{ defined by }\ w \mapsto uv\ \text{ and }\ z
\mapsto 1/v\\ \intertext{and}
g_{10} &: V_{10} \to V_{01}\ \text{ coming from 
the $\C$-algebra homomorphism }\\ g_{10}^* &: \C[u,v]_v
\to \C[w,z]_z \ \text{ defined by }\ u \mapsto wz\ \text{ and }\ v 
\mapsto 1/z. 
\end{align*}
One checks that $g_{01} = g_{10}^{-1}$, and the other compatibility
condition is satisfied since there are only two $V_i$.  It follows
that we get an abstract variety $X$.

The variety $X$ has another description.  Consider the
product $\P^1 \times \C^2$ with homogeneous coordinates $(x_0,x_1)$ on
$\P^1$ and coordinates $(x,y)$ on $\C^2$.  We will identify $X$ with
the subvariety $W = \V(x_0y - x_1 x) \subseteq \P^1 \times \C^2$,
called the \emph{blow-up of $\C^2$ at the origin}, and denoted
$\mathrm{Bl}_0(\C^2)$.  First note that $\P^1 \times \C^2$ is covered by 
\[
U_0 \times \C^2 = \Spec(\C[x_1/x_0,x,y]) \quad\text{and}\quad
U_1 \times \C^2 = \Spec(\C[x_0/x_1,x,y]).
\]
Then $W$ is covered by $W_0 = W \cap (U_0 \times \C^2)$ and $W_1 = W
\cap (U_1 \times \C^2)$.  Also,
\[
W_0 = \V(y - (x_1/x_0)x) \subseteq U_0 \times \C^2,
\]
which gives the coordinate ring
\[
\C[x_1/x_0,x,y]/\langle y - (x_1/x_0)x \rangle \simeq \C[x,x_1/x_0]
\qquad \text{via } y \mapsto (x_1/x_0)x.
\]
Similarly, $W_1 = \V(x - (x_0/x_1)y) \subseteq U_1 \times \C^2$ has
coordinate ring
\[
\C[x_0/x_1,x,y]/\langle x - (x_0/x_1)y \rangle \simeq \C[y,x_0/x_1]
\qquad \text{via } x \mapsto (x_0/x_1)y.
\]
You should check that these are glued together in $W$ in exactly the
same way that $V_0$ and $V_1$ are glued together in $X$. 
We will generalize this example in Exercise~\ref{3.blowups}.\eoe
\end{example}

{}From now on, when considering varieties, we will usually be thinking
of abstract varieties.  Hence we will usually say ``variety'' rather
than ``abstract variety.''  

\subsection*{\emph{The Local Rings of a Variety}}
The definition of local ring of an irreducible affine variety extends
easily to the abstract case.  A \emph{rational function} on an
irreducible variety $X$ is a regular function $f : U \to \C$
defined on a nonempty Zariski open set of $X$, and two rational
functions on $X$ are \emph{equivalent} if they agree on a nonempty
Zariski open subset.  In Exercise~\ref{3.funcfieldprops} you will show
that this relation is an equivalence relation and that the set of
equivalence classes is a field, called the \emph{function field} of
$X$, denoted $\C(X)$.

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 $p$ be a point of an irreducible variety $X$.  Then
\[
\cO_{X,p} = \{ \phi \in \C(X) \mid \phi \text{ is defined at } p\}.   
\]
is the \emph{\bfseries local ring of $X$ at $p$}.
\end{definition}

It is not difficult to see that $\cO_{X,p}$ is a local ring with
unique maximal ideal
\[
\mf_{X,p} = \{\phi \in \cO_{X,p}\ |\ \phi(p) = 0\}.  
\]
The local ring $\cO_{X,p}$ can also be defined for general (possibly
reducible) varieties as the direct limit
\[
\cO_{X,p} = 
\lim_{\stackrel{\scriptstyle \longrightarrow}{\scriptstyle p\in U}}
\cO_X(U), 
\]
over all neighborhoods of $p$ in $X$.  See \cite{eh} or 
\cite{hartshorne}.

\subsection*{\emph{Smooth Varieties and Orbifolds}}

Because the definition of \emph{smoothness} at a point $p$ for 
an affine variety $V$ depends only on the local ring $\cO_{V,p}$ 
(see Definition~\ref{1.smoothdef} from Chapter~\ref{affine}), we 
can extend this notion to general varieties using the local rings 
as defined above (see Exercise~\ref{3.welldef}). 

\begin{definition} 
\label{3.smoothorb}
Let $X$ be a variety. We say  $X$ is \emph{\bfseries smooth}
if every point $p\in X$ has an affine
open neighborhood that is a smooth affine variety.
\end{definition}

A similar comment lets us extend the notion of 
\emph{finite quotient singularities}
to a general variety and to define \emph{orbifolds} 
in this more general context.   

\subsection*{\emph{Morphisms of Varieties}} Given varieties $X$ and
$Y$ with affine open covers $X = \bigcup_\a U_\a$ and $Y = \bigcup_\b
U'_\b$, a function $\Phi : X \to Y$ is a \emph{morphism} if it is
continuous in the Zariski topology and the restrictions
\[
\Phi\res{U_\a \cap \Phi^{-1}(U'_\b)} : U_\a \cap \Phi^{-1}(U'_\b)
\longrightarrow U'_\b
\]
are morphisms in the sense of Definition~\ref{3.mordef1}.  One can
check that this definition is indpendent of the affine open covers.
Also, a function $\phi : X \to \C$ is a morphism if and only if it
gives an element of $\cO_X(X)$, i.e., $\phi$ is  a regular function.

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

\begin{definition}
\label{3.normalvardef}
An irreducible variety $X$ is called \emph{\bfseries normal} if the
local rings $\cO_{X,p}$ are normal for all $p \in X$.
\end{definition}

At first glance, this appears to be different from the definition
given for affine varieties in Definition~\ref{1.normaldef} of
Chapter~\ref{affine}.  In fact the two notion are equivalent in the
affine case.

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

\begin{proof}
If $\cO_{V,p}$ is normal for all $p$, then \eqref{3.intlocalrings}
shows that $\C[V]$ is an intersection of normal domains, all of which
have the same field of fractions.  Since such an intersection is
normal by Exercise~\ref{1.0.5} of Chapter~\ref{affine}, it follows
that $\C[V]$ is normal.  

For the converse, suppose that $\C[V]$ is normal and let $\a \in
\C(V)$ satisfy
\[
\a^k + a_1 \a^{k-1} + \cdots + a_k = 0,\qquad a_i \in \cO_{V,p}.
\]
Write $a_i = g_i/f_i$ with $g_i,f_i \in \C[V]$ and $f_i(p) \ne 0$.
The product $f = f_1\cdots f_k$ has the properties that $a_i \in
\C[V]_f$ and $f(p) \ne 0$.  The localization $\C[V]_f$ is normal by
Exercise~\ref{1.0.5} of Chapter~\ref{affine} and is contained in
$\cO_{V,p}$ since $f(p) \ne 0$.  Hence $\a \in \C[V]_f \subseteq
\cO_{V,p}$.  This completes the proof.
\end{proof}

We have the following 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{Products of Varieties}}
As another example of abstract varieties and gluing, we indicate why
the product $X_1\times X_2$ of varieties $X_1$ and $X_2$ also has the
structure of a variety.  In \S0 of Chapter~\ref{affine} we constructed
the product of affine varieties.  {}From here, it is relatively routine
to see that if $X_1$ is obtained by gluing together affine varieties
$U_\a$ and $X_2$ is obtained by gluing together affines $U'_\b$, then
$X_1 \times X_2$ is obtained by gluing together the $U_\a \times U'_\b$
in the corresponding fashion.  Furthermore, $X_1\times X_2$ has the
correct universal mapping property.  Namely, given a diagram
\[
\SelectTips {cm}{}
\xymatrix{
W \ar@/^1pc/[drr]^{\phi_1} \ar@/_1pc/[ddr]_{\phi_2} \ar@{.>}[dr]|{\nu} & & \\
& X_1 \times X_2 \ar[r]_(.6){\pi_1} \ar[d]^{\pi_2} & X_1\\
& X_2 &
}
\]
where $\phi_i : W \to X_i$ are morphisms, there is a unique morphism
$\nu: W \to X_1 \times X_2$ (the dotted arrow) that makes the diagram
commute.

The details of products and their properties are presented in
\cite{eh} and \cite{hartshorne}.

\begin{example}
\label{3.p1c2}
Let us construct the product $\P^1 \times \C^2$.  Write $\P^1 = V_0
\cup V_1$ where $V_0 = \Spec(\C[u])$ and $V_1 = \Spec(\C[v])$, with
the gluing given by
\[
\C[v]_{v} \simeq \C[u]_{u}, \qquad v \mapsto 1/u.
\]
Then $\P^1 \times \C^2$ is constructed from 
\begin{align*}
U_0 \times \C^2 &= \Spec(\C[u]\otimes\C[x,y] ) \simeq \C^3\\
U_1 \times \C^2 &= \Spec(\C[v]\otimes\C[x,y] ) \simeq \C^3,
\end{align*}
with gluing given by 
\[
(U_0 \times \C^2)_{u} \simeq (U_1 \times \C^2)_{v}
\]
corresponding to the obvious isomorphism of coordinate rings. \eoe
\end{example}

\subsection*{\emph{Separated Varieties}}
{}From the point of view of the classical topology, arbitrary gluings
can lead to varieties with some strange properties.

\begin{example}
\label{3.nonsep}
In Example~\ref{3.p1c2} we saw how to construct $\P^1$ from affine
varieties $V_0 = \Spec(\C[u]) \simeq \C$ and $V_1 = \Spec(\C[v])
\simeq \C$ with the gluing given by $v \mapsto 1/u$ on open sets
$\Cstar \simeq (V_0)_u \subseteq V_0$ and $\Cstar \simeq (V_1)_v
\subseteq V_1$.  This expresses $\P^1$ as consisting of $\Cstar$ plus
two additional points.  But now consider the abstract variety arising
from the gluing map 
\[
(V_0)_u \longrightarrow (V_1)_v
\]
that corresponds to the map of $\C$-algebras defined by $v \mapsto u$.
As before, 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 $X$ is not Hausdorff.  \eoe
\end{example}

Since varieties are rarely Hausdorff in the Zariski topology
(Exercise~\ref{3.ZariskiHausdorff}), we need a different way to
understand Example~\ref{3.nonsep}.  Consider the product $X\times X$
and the \emph{diagonal mapping} $\Delta : X \to X \times X$ defined by
$\Delta(p) = (p,p)$ for $p\in X$.  For $X$ from
Example~\ref{3.nonsep}, there is a morphism $X\times X \to \C$ whose 
fiber over over $0$ consists of the four points $(p_i,p_j)$.  Any Zariski
closed subset of $X \times X$ containing one of these four points must
contain all of them.  The image of the diagonal mapping contains
$(p_1,p_1)$ and $(p_2,p_2)$, but not the other two, so the diagonal is
not Zariski closed.  This example motivates the following definition.

\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 $\V(x_1 - y_1, \ldots, x_n - y_n)$.  Similarly
any affine variety is separated.

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

\begin{theorem}  
\label{3.sepHaus}
A variety is separated if and only if
it is Hausdorff in the classical topology.  \qed
\end{theorem}

Here are some additional properties of separated varieties
(Exercise~\ref{3.seppropsex}).

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

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

\subsection*{\emph{Fibered Products}}
Finally in this section, we will discuss fibered products of varieties,
a construction required for the discussion of proper morphisms
in \S 4.  First, if we have mappings of sets $f: X \to S$ and 
$g : Y \to S$, then the fibered product $X \times_S Y$ is defined
to be
\[
X \times_S Y = \{(x,y) \in X \times Y \mid f(x) = g(y)\}.
\]
The fibered product construction gives a very flexible language
for describing ordinary products, intersections of subsets, 
fibers of mappings, the set where two mappings agree and so forth:
\begin{coxitem}
\item  If $S$ is a point, then $X \times_S Y$ is the ordinary
product $X\times Y$.
\item If $X$ and $Y$ are subsets of $S$ and $f,g$ are the 
inclusion mappings then $X\times_S Y = X\cap Y$.
\item If $Y = \{s\} \subseteq S$, then $X \times_S Y = f^{-1}(\{s\})$.
\item If $X = Y$, then $X \times_S X = \{(x,x') \mid f(x) = g(x')\}$,
(compare with Proposition~\ref{3.sepprops}).
\end{coxitem}
This third property is the reason for the name.  All are easy 
exercises that we leave for the reader.  

In analogy with the universal mapping property of the product
discussed above, we have the following statement.  Whenever
we have mappings $\phi_1 : W \to X$ and $\phi_2 : W \to Y$
such that $f\circ \phi_1 = g \circ \phi_2$, there is a unique
$\nu : W \to X \times_S Y$ that makes the following diagram
commute.
\[
\SelectTips {cm}{}
\xymatrix{
W \ar@/^1pc/[drr]^{\phi_1} \ar@/_1pc/[ddr]_{\phi_2} \ar@{.>}[dr]|{\nu} & & \\
& X \times_S Y \ar[r]_(.6){\pi_1} \ar[d]^{\pi_2} & X \ar[d]^{f} \\
& Y \ar[r]_(.6){g} & S
}
\]
If $X = \Spec(R_1)$, $Y = \Spec(R_2)$, and $S = \Spec(R)$ are 
affine varieties, then the fibered product also exists as
an affine variety.  Namely, the morphisms $f,g$ correspond
to ring homomorphisms
\begin{eqnarray*}
f^* : R \to R_1\\
g^* : R \to R_2
\end{eqnarray*}
Hence both $R_1, R_2$ have the structure of $R$-modules, and
we have the tensor product $R_1 \otimes_R R_2$.  This is 
also a finitely generated $\C$-algebra without nilpotents, so it is the 
coordinate ring of an affine variety.   So we take 
\[
X \times_S Y = \Spec(R_1 \otimes_R R_2)
\]  
as the definition of $X \times_S Y$.  When $S,X,Y$ are general
varieties, $X\times_S Y$ is constructed by the gluing procedure.
We will not present the details here; interested readers should
consult the discussion of the general construction within 
the theory of schemes given in \cite{eh,hartshorne}.

\exercises

\small

\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.  (This says that affine varieties are
\emph{quasicompact} in the Zariski topology.)
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{3.welldef}
As in the affine case, we want to say a variety $X$ is smooth at $p$ 
if $\dim T_p(X) = \dim X$.  In this exercise, you will show that 
this is a well-defined notion.
\begin{thmenum}
\item Show that if $p\in X$ is 
in the intersection of two affine open sets $V_\a \cap V_\b$, then
the Zariski tangent spaces $T_{V_\a,p}$ and $T_{V_\b,p}$ are
isomorphic as vector spaces over $\C$.
\item Show that $\dim X$ is a well-defined integer.
\item Deduce that the proposed notion of smoothness at $p$
is well-defined.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{3.morphismexer}
This exercises explores some properties of the morphisms described in
Definition~\ref{3.mordef1}. 
\begin{thmenum}
\item Prove the claim made in Example~\ref{3.morphismex}.  Hint:\ Take
$p \in V_1$ and set $\mf_p = \{f \in R_1 \mid f(p) = 0\}$.  Then
describe $(\Phi^*)^{-1}(\mf_p)$ in terms of $\Phi(p)$.
\item Prove the properties of morphisms listed on page
\pageref{3.morphismprops}.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{3.funcfieldprops}
Let $X$ be an irreducible abstract variety.
\begin{thmenum}
\item Let $f,g$ be rational functions on $X$.  Show that $f \sim g$ if
$f|_U = g|_U$ for some nonempty open set $U \subseteq X$ is an
equivalence relation.
\item Show that the set of equivalence classes of the relation in part
(a) is 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.ZariskiHausdorff}
Show that a variety is Hausdorff in the Zariski topology if and
only if it is a finite set.
\end{exercise}

\begin{exercise}
\label{3.seppropsex}
Consider Proposition~\ref{3.sepprops}.
\begin{thmenum}
\item Prove part (a) of the proposition.  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.  Hint:\ Show first that $U \cap
V$ is identified with $\Delta(X) \cap (U\times V) \subseteq X\times
X$.
\end{thmenum}
\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.blowups}
In this exercise, we will study an important variety in $\P^{n-1} \times \C^n$,
the \emph{blow-up} of $\C^n$ at the origin, denoted $\mathrm{Bl}_0(\C^n)$.  
This generalizes $\mathrm{Bl}_0(\C^2)$ from Example~\ref{3.blowuppt}.
Write the homogeneous coordinates on $\P^{n-1}$ as $(x_0,\ldots,x_{n-1})$,
and the affine coordinates on $\C^n$ as $(y_1,\ldots,y_n)$.  Let
\begin{equation}
\label{3.blowupeqs}
W = \mathrm{Bl}_0(\C^n) = \V(x_{i-1}y_j - x_{j-1}y_i \mid 1 \le i < j \le n) \subseteq \P^{n-1}\times \C^n.
\end{equation}
Let $U_{i-1}$, $i = 1, \ldots, n$ be the standard affine opens in $\P^{n-1}$:
\[
U_{i-1} = \P^{n-1} \setminus \V(x_{i-1}),
\]
$i = 1, \ldots, n$ (note the slightly non-standard indexing).
So the $U_{i-1} \times \C^n$ form a cover of $\P^{n-1}\times \C^n$.
\begin{thmenum}
\item Show that for each $i = 1, \ldots, n$,
$W_{i-1} = W \cap (U_{i-1} \times \C^n) \simeq$
\[
\Spec\left(\C\left[\frac{x_0}{x_{i-1}},\ldots,\frac{x_{i-2}}{x_{i-1}},
\frac{x_{i}}{x_{i-1}},\ldots,\frac{x_{n-1}}{x_{i-1}},y_i\right]\right)
\]
using the equations \eqref{3.blowupeqs} defining $W$.
\item Give the gluing data for identifying the subsets
$W_{i-1} \setminus \V(x_{j-1})$ and $W_{j-1} \setminus \V(x_{i-1})$.  
\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 see how properties of
the fan correspond to properties such as smoothness and compactness of 
$X_\Sigma$.  

\subsection*{\emph{The Toric Variety Associated to a Fan}}
A toric variety in this context means the same thing as in
Chapters 1 and 2, although we now allow abstract varieties as in
\S 0.  

\begin{definition}
\label{3.abstoric}
A \emph{\bfseries 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}

The other ingredient in this section is a fan in the
vector space $N_\R$.  

\begin{definition}
\label{3.fandef}
A \emph{\bfseries fan} $\Sigma$ in $N_\R$ is a finite 
collection of cones $\sigma$ such that
\begin{thmenum}
\item every $\sigma$ in $\Sigma$ is a strongly convex
rational polyhedral cone,
\item for all $\sigma$ in $\Sigma$, each face of $\sigma$ is also
in $\Sigma$, and  
\item if $\sigma_1$ and $\sigma_2$ are in $\Sigma$,
then the intersection $\sigma_1\cap\sigma_2$ is a face of each
(hence also in $\Sigma$).
\end{thmenum}
\end{definition}

It is also possible to consider infinite collections of 
cones with the same properties, but we will not do so
in this book.  

Examples of fans include the normal fans of lattice
polytopes $P \subseteq M_\R$ as in Chapter~\ref{poly}, \S 3,
and the properties given in this definition were
verified for those fans in Corollary~\ref{2.norfancombcor} in that section.

However, there exist fans that are not equal to the
normal fan of any lattice polytope.  
The construction of such fans will be easier to motivate
after we have discussed the connections between polytopes, 
divisors, and rational mappings from toric varieties to $\P^n$.  
We will postpone discussing them until Chapter~\ref{invertible}.

We now show how the cones in any fan give the combinatorial
data necessary to glue a collection of affine toric varieties
together to yield an abstract toric variety.  
By Chapter~\ref{affine}, Theorem~\ref{1.coneafftor},
for each cone $\sigma$ in $\Sigma$, we have the affine toric variety 
\[
U_\sigma = \Spec(\C[\sS_\sigma]) = \Spec(\C[\sigma^\vee \cap M]).
\]
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 defined by $m$.  In Proposition~\ref{1.facesg} of Chapter~1,
we proved the following facts:
\begin{coxitem}
\item If $m \in M$ and $\tau = \sigma \cap H_m$ is the corresponding
face of $\sigma$, then 
\begin{equation}
\label{3.face}
\sS_\tau = \sS_\sigma + \Z (-m),
\end{equation}
and $\C[\sS_\tau]$ is the localization $\C[\sS_\sigma]_{\chi^m}$.
Therefore, $U_\tau = (U_\sigma)_{\chi^m}$.
\item If $\tau = \sigma_1\cap \sigma_2$, then there exists 
$m \in \sigma_1^\vee \cap (-\sigma_2)^\vee \cap M$
such that 
\[
\sigma_1\cap H_m = \tau = \sigma_2\cap H_m
\]
and
\begin{equation}
\label{3.overlap}
U_{\sigma_1} \supseteq (U_{\sigma_1})_{\chi^m} = U_\tau = 
(U_{\sigma_2})_{\chi^{-m}} \subseteq U_{\sigma_2}.
\end{equation}
\end{coxitem}

These facts were also used to study projective toric varieties
in Chapter~\ref{poly}.
The following proposition gives an additional property
of the $\sS_\sigma$ and their semigroup rings that we will need
in this section.  

\begin{proposition}
\label{3.semiprops}
If $\sigma_1,\sigma_2$ are rational polyhedral cones such
that $\tau = \sigma_1 \cap \sigma_2$ is a face of each $\sigma_i$,
then 
\[
\sS_\tau = \sS_{\sigma_1} + \sS_{\sigma_2}.
\]
\end{proposition}

\begin{proof}
The inclusion $\sS_{\sigma_1} + \sS_{\sigma_2} \subseteq \sS_\tau$
follows directly from the general fact that
$\sigma_1^\vee + \sigma_2^\vee  = (\sigma_1 \cap \sigma_2)^\vee = \tau^\vee$.
The reverse inclusion then follows from the bulleted points above.  
If $p \in \sS_\tau$, then by \eqref{3.face}, 
$p = q + \ell (-m)$ for some $q \in \sS_{\sigma_1}$ and $\ell\in \Z_{\ge 0}$. Since 
$-m \in (-\sigma_2)^\vee$ implies $m \in \sS_{\sigma_2}$, we have
$p \in \sS_{\sigma_1} + \sS_{\sigma_2}$.
\end{proof}

This result is sometimes called the \emph{separation lemma}, and it will
be the key ingredient in showing that the toric varieties $X_\Sigma$
are separated using Definition~\ref{3.sepdef}.

\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 $N_\R \simeq \R^2$.  
\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.8in]{ch3fig1.ps}
\end{center}
\caption{The cones $\sigma_1,\sigma_2,\tau$ and their duals} 
\end{figure}
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 Figure~1.

The darkest shaded region on the right
is $\sigma_1^\vee \cap \sigma_2^\vee$.  
We have $\tau = \sigma_1 \cap H_m = \sigma_2 \cap H_{-m}$, 
where $m = -e_1 + e_2 \in \sigma_1^\vee$
and $-m = e_1-e_2 \in \sigma_2^\vee$.   
Note that $\sS_\tau$ equals the set of
all sums $m + m'$ with $m \in \sigma_1^\vee \cap M$ and $m' \in \sigma_2^\vee\cap M$.
Hence $\sS_\tau = \sS_{\sigma_1} + \sS_{\sigma_2}$.
\eoe
\end{example}
 

Now consider the collection of affine
toric varieties $U_\sigma = \Spec(\C[\sS_\sigma])$, where
$\sigma$ runs over all cones in a fan $\Sigma$.  Let $\sigma_1$ and
$\sigma_2$ be any two of these cones and let $\tau = \sigma_1 \cap \sigma_2$.
By \eqref{3.overlap}, we have
an isomorphisms 
\[
g_{\sigma_1,\sigma_2}: (U_{\sigma_1})_{\chi^m} \simeq (U_{\sigma_2})_{\chi^{-m}}.
\]
(In fact
this mapping is just the identity on $U_\tau$.)
In Exercise~\ref{3.fangluing}, you will show that the compatibility conditions as 
in \S 0 for gluing the affine varieties $U_\sigma$ along the subvarieties 
$(U_{\sigma})_{\chi^m}$ are satisfied.
Hence, we obtain an abstract variety $X_\Sigma$ associated to the fan $\Sigma$.

\begin{theorem}
\label{3.toricfromfan}
Let $\Sigma$ be a fan in $N_\R$.  
The variety $X_\Sigma$ is a normal, separated 
toric variety.
\end{theorem}

\begin{proof}
First, $X_\Sigma$ is irreducible because all of the $U_\sigma$ are
irreducible affine toric varieties.

Since each cone in a fan is strongly convex, $\{0\}\subseteq N$ is a face of all
$\sigma \in \Sigma$.  Hence we have $T_N = \Spec(\C[M]) \simeq (\Cstar)^n \subseteq U_\sigma$
for all $\sigma$.  These tori are all identified by the gluing, so we have
$T_N \subseteq X_\Sigma$.  We know from Chapter~\ref{affine} that
each $U_\sigma$ has an action of $T_N$.  The gluing isomorphism
$g_{\sigma_1,\sigma_2}$ reduces to the identity mapping on 
$\C[\sS_{\sigma_1\cap\sigma_2}]$.  Hence the actions are compatible
on the intersections of every pair of sets in the open affine cover, and patch
together to give an action of $T_N$ on $X_\Sigma$.
 
By Theorem~\ref{1.satthm} of Chapter~\ref{affine}, for each $\sigma \subseteq \Sigma$,
$U_\sigma$ 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 
\[
\Delta : U_{\tau} \to U_{\sigma_1} \times U_{\sigma_2}
\]
is Zariski closed (Exercise~\ref{3.sepcover}). 
But $\Delta$ comes from the $\C$-algebra homomorphism 
\[
\Delta^* : \C[\sS_{\sigma_1}] \otimes_\C \C[\sS_{\sigma_2}] \longrightarrow \C[\sS_\tau]
\]
defined by $\chi^m \otimes \chi^n \mapsto \chi^{m+n}$.  By 
Proposition~\ref{3.semiprops}, $\Delta^*$ is surjective, which shows
that 
\[
\C[\sS_\tau]\simeq (\C[\sS_{\sigma_1}] \otimes_\C \C[\sS_{\sigma_2}])/I
\]
for some ideal $I$.  Hence
the image of $\Delta$ is a Zariski closed subset of $V_{\sigma_1} \times V_{\sigma_2}$.
\end{proof}

Toric varieties were originally known as \emph{torus embeddings}, and the 
variety $X_\Sigma$ would be written $T_N\mathrm{emb}(\Sigma)$ in older references
such as \cite{oda}.  Other commonly used notations are $X(\Sigma)$, or 
$X(\Delta)$, if the fan is denoted by $\Delta$.

\emph{Every} normal, separated
toric variety is obtained as $X_\Sigma$ for some fan $\Sigma$ in $N_\R$.
For example, Theorem~\ref{1.satthm} of Chapter~\ref{affine} implies that
every affine toric variety is $X_\Sigma$ for the fan consisting of a 
single cone $\sigma$ together with all of its faces.
The construction of the projective toric variety associated
to a lattice polytope in Chapter~\ref{poly} shows that those
varieties are also $X_\Sigma$ for suitable fans $\Sigma$.

\begin{proposition}
\label{3.polynormalfan}
Let $P \subset M_\R \simeq \R^n$ be an $n$-dimensional lattice
polytope.  Then the projective toric variety $X_P \simeq X_{\Sigma_P}$, 
where $\Sigma_P$ is the normal fan of $P$.
\end{proposition}

\begin{proof}
When $P$ is very ample, this follows immediately from 
the description of the intersections of the affine open
pieces of $X_P$ in Proposition~\ref{2.uvuw} from Chapter~\ref{poly}
and the definition of the normal fan $\Sigma_P$.  The general
case follows since the normal fans of $P$ and $kP$ are the same
for all positive integers $k$.
\end{proof}

The general statement is a consequence of a theorem of Sumihiro 
from \cite{sumihiro}.

\begin{theorem}[Sumihiro]
\label{3.sumihiro}
Let the torus $T_N$ act on an irreducible, normal, separated variety $X$.
Then every point $p \in X$ has a $T_N$-invariant affine open neighborhood. \qed
\end{theorem}

\begin{corollary}
\label{3.allfromfans}
Let $X$ be an irreducible, normal, separated toric variety $X$ containing
the torus $T_N$ as an affine open subset.
Then there exists a fan $\Sigma$ in $N_\R$ such that $X \simeq X_\Sigma$.
\end{corollary}

The proof of Corollary~\ref{3.allfromfans}, assuming Theorem~\ref{3.sumihiro},
will be sketched in Exercise~\ref{3.Cor17} in \S 2 after we have developed some of the 
properties of $T_N$-orbits on toric varieties.  

\subsection*{\emph{Additional Examples}}
We now turn to some concrete examples.  Many of these are toric varieties
already encountered in previous chapters.   

\begin{example}
\label{3.P2}

Consider the fan $\Sigma$ in $N_\R \simeq \R^2$ in Figure~2.
This is the same as the normal fan of each of the simplices $P = k\Delta_2$
as in Example~\ref{2.p2ex} from Chapter~\ref{poly}.  Here we show
all points in the cones inside a rectangular viewing box
(all figures of fans in the plane in this chapter will be
drawn using the same convention.)
\begin{figure}[ht]
\begin{center}
\includegraphics[height=2.0in]{ch3fig2.ps}
\end{center}
\caption{The fan $\Sigma$ for $\P^2$} 
\end{figure}

From the discussion in Chapter 2, we expect $X_\Sigma \simeq \P^2$, 
and we will show this in detail.  The fan $\Sigma$ has three two-dimensional
cones $\sigma_0 = \Cone(e_1,e_2)$, $\sigma_1 = \Cone(-e_1-e_2,e_2)$, and
$\sigma_2 = \Cone(e_1,-e_1-e_2)$, together with the three rays 
$\tau_{ij} =\sigma_i \cap \sigma_j$
for $i \ne j$, and the origin.
We see that the toric variety 
$X_\Sigma$ is covered by the affine opens
\begin{eqnarray*}
U_{\sigma_0} &=& \Spec(\C[\sS_{\sigma_0}]) \simeq \Spec(\C[x,y]),\\
U_{\sigma_1} &=& \Spec(\C[\sS_{\sigma_1}]) \simeq \Spec(\C[x^{-1},x^{-1}y]),\\
U_{\sigma_2} &=& \Spec(\C[\sS_{\sigma_2}]) \simeq \Spec(\C[xy^{-1},y^{-1}]).
\end{eqnarray*}
Moreover, by Proposition~\ref{3.semiprops}, the gluing data on the 
coordinate rings is given by the isomorphisms
\begin{eqnarray*}
g_{10}^* : \C[x,y]_x &\simeq& \C[x^{-1},x^{-1}y]_{x^{-1}},\\
g_{20}^* : \C[x,y]_y &\simeq& \C[xy^{-1},y^{-1}]_{y^{-1}},\\
g_{21}^* : \C[x^{-1},x^{-1}y]_{x^{-1}y} &\simeq& \C[x,xy^{-1}]_{xy^{-1}}.
\end{eqnarray*}
It is easy to see that if we use the usual homogeneous coordinates
$(x_0,x_1,x_2)$ on $\P^2$, then $x \mapsto \frac{x_1}{x_0}$ and $y \mapsto \frac{x_2}{x_0}$
identifies the standard affine open $U_i$ with $U_{\sigma_i}$.  Hence we 
have recovered $\P^2$ as the toric variety $X_\Sigma$.  
\eoe
\end{example}

\begin{example}
\label{3.Pn}
Generalizing Example~\ref{3.P2}, let $N_\R \simeq \R^n$, and write 
\[
e_0 = -e_1 - e_2 - \cdots - e_n.
\]  
Let $\Sigma$ be the 
fan in $N_\R$ consisting of the cones generated by all 
proper subsets of $\{e_0,\ldots,e_n\}$.
This is the normal fan of the $n$-simplex $\Delta_n$ and we know
the toric variety $X_\Sigma \simeq \P^n$ by Example~\ref{2.veronese}
and Exercise~\ref{2.3.veroneseexer} of Chapter~\ref{poly}.  
(You will check the details to verify that the general gluing 
construction gives the usual affine open cover of $\P^n$ in 
Exercise~\ref{3.Pnex}).   To relate this example to 
the discussion in Chapter~\ref{poly}, note that we have different 
embeddings of $\P^n$ according to which $k \Delta_n$ is
used to construct the Veronese mapping.  However the normal
fans of $k\Delta_n$ are all the same as this fan, and the
varieties obtained this way for different $k$ are isomorphic.  
\eoe
\end{example}

\begin{example}
\label{3.PnPm}
Let $N = N_1 \times N_2$ with $(N_1)_\R \simeq \R^n$ and $(N_2)_\R \simeq \R^m$.
Let $\Sigma_i$ be the fan in $(N_i)_\R$ as in Example~\ref{3.Pn}.  Then
$\Sigma = \Sigma_1 \times \Sigma_2$ is a fan in $N$, consisting of all
cones $\sigma_1 \times \sigma_2$, where $\sigma_i$ is in $\Sigma_i$.  
By Example~\ref{2.prodex} of Chapter~\ref{poly}, the corresponding toric variety
is the product $\P^n \times \P^m$.  For instance, if $n = m = 1$, we obtain
the fan $\Sigma \subseteq \R^2 \simeq N_\R$ pictured in Figure~3.
\begin{figure}[ht]
\begin{center}
\includegraphics[height=2.0in]{ch3fig3.ps}
\end{center}
\caption{A fan $\Sigma$ with $X_\Sigma \simeq \P^1 \times \P^1$} 
\end{figure}

This example was also studied in Example~\ref{2.P1P1} in Chapter~\ref{poly}.
Label the 2-dimensional cones $\sigma_{ij} = \sigma_i \times \sigma'_j$
as above.  Then 
\begin{eqnarray*} 
\Spec(\C[\sS_{\sigma_{00}}]) &\simeq& \C[x,y],\\
\Spec(\C[\sS_{\sigma_{10}}]) &\simeq& \C[x^{-1},y],\\
\Spec(\C[\sS_{\sigma_{11}}]) &\simeq& \C[x^{-1},y^{-1}],\\
\Spec(\C[\sS_{\sigma_{01}}]) &\simeq& \C[x,y^{-1}].
\end{eqnarray*}
We see that if $U_0$ and $U_1$ are the standard affine open sets in $\P^1$,
then $U_{\sigma_{ij}} \simeq U_i \times U_j$ and it is easy to check
that the gluing makes $X_\Sigma \simeq \P^1 \times \P^1$.  
\eoe
\end{example}

\begin{example}
\label{3.PnCm}
Let $N = N_1 \times N_2$, with $N_i$ as in the previous example.
Let $\Sigma_1$ in $(N_1)_\R$ be as above, but
let $\Sigma_2$ be the fan consisting of the cone $\Cone(e_1,\ldots,e_n)$
together with all its faces.  Then $\Sigma = \Sigma_1 \times \Sigma_2$
is a fan in $N_\R$ and the 
the corresponding toric variety is $X_{\Sigma} \simeq \P^m\times \C^n$.
The case $\P^1\times \C^2$ was studied in Example~\ref{3.p1c2}.
\eoe
\end{example}

\begin{example}
\label{3.faninsubspace}
Let $N \subseteq N'$ be a sublattice.  If $\Sigma$ is a fan
in $N_\R$, then we can also view it as a fan in $(N')_\R$.  What 
is the relation between $X_{\Sigma,N}$ and $X_{\Sigma,N'}$?
We will study this in much more detail in \S 3 of this Chapter.
Here is a simple example showing one possibility.  Let $\Sigma$
be the fan pictured below, but viewed as a fan in $(N')_\R \simeq \R^2$.
\[
\begin{picture}(220,25)
\thicklines
\put(10,15){\line(1,0){200}}
\put(110,15){\circle*{4}}
\put(108,4){$0$}
\put(60,5){$\sigma_1$}
\put(160,5){$\sigma_0$}
\end{picture}
\]
It is easy to check that the corresponding $X_{\Sigma,N'} \simeq \P^1 \times \Cstar$
(Exercise~\ref{3.P1Cstar}).  In general, when $N$ is spanned
by a subset of a basis for $N'$, we will 
show in \S 3 that $X_{\Sigma,N'} \simeq X_\Sigma \times T_{N'/N}$.
\eoe
\end{example}

\begin{example}
\label{3.blowupptfan}
The two cones $\sigma_1$ and $\sigma_2$ in $N_\R \simeq \R^2$
from Example~\ref{3.semipropsex} (see Figure~1), together with their faces,
form a fan $\Sigma$.  By comparing the descriptions of the 
coordinate rings of $V_{\sigma_i}$ given there with 
what we did in Example~\ref{3.blowuppt}, it is easy to check
that $X_\Sigma \simeq W$, where $W \subseteq \P^1 \times \C^2$
is the blowup of $\C^2$ at the origin, defined as $W = \V(x_0y - x_1x)$
(Exercise~\ref{3.blowuporigin}).

Generalizing this,  
let $N \simeq \Z^n$, let $e_1,\ldots,e_n \in N$ be the standard
basis and let $e_0 = e_1 + \cdots + e_n$.  Let $\Sigma$ be the fan
in $N_\R$ consisting of the cones generated by all subsets of 
$\{e_0,\ldots,e_n\}$ \emph{not containing} $\{e_1,\ldots,e_n\}$.
Then the toric variety $X_\Sigma$ is isomorphic to the blow-up 
of $\C^n$ at the origin (see Exercise~\ref{3.blowups}).
\eoe
\end{example}

\begin{example}
\label{3.hirzebruch}
Let $r \in \Z_{\ge 0}$ and consider the fan $\Sigma$ 
in $N_\R \simeq \R^2$ consisting of the four cones $\sigma_i$
shown in Figure~4, together
with all of their faces. 
\begin{figure}[ht]
\begin{center}
\includegraphics[height=2.0in]{ch3fig4.ps}
\end{center}
\caption{A fan $\Sigma$ with $X_\Sigma \simeq \cH_r$} 
\end{figure}
The corresponding
variety $X_\Sigma$ is covered by open affines, each isomorphic to $\C^2$,
\begin{eqnarray*}
U_{\sigma_1} &=& \Spec(\C[x,y])\\
U_{\sigma_2} &=& \Spec(\C[x,y^{-1}])\\
U_{\sigma_3} &=& \Spec(\C[x^{-1},x^{-r}y^{-1}])\\
U_{\sigma_4} &=& \Spec(\C[x^{-1},x^r y]),
\end{eqnarray*}
and glued according to the recipe from \eqref{3.overlap}.
The resulting $X_\Sigma$ is called the \emph{Hirzebruch surface}
$\cH_r$.  We saw a projective toric variety, the \emph{rational normal
scroll} $S_{2,4}$, defined by a polytope $P$ with normal fan
equal to $\Sigma$ as here with $r = 2$ in Example~\ref{2.rnscroll}
in Chapter~\ref{poly}.  Note also that $\cH_0 \simeq \P^1\times \P^1$. 
We will study these varieties in detail in Chapter~\ref{surface}.
\eoe
\end{example}

\begin{example}
\label{3.weightedproj}
Let $q_0,\ldots,q_n \in \Z_{\ge 0}$ satisfy $\gcd(q_0,\ldots,q_1) = 1$.
Consider the weighted projective space $\P(q_0,\ldots,q_n)$ introduced
in Chapter~\ref{poly}.  Let $N = \Z^{n+1}/\Z\cdot(q_0,\ldots,q_n)$
and let $v_i$, $i = 0,\ldots,n$ be the images in $N$ of the standard basis vectors
in $\Z^{n+1}$, so the relation 
\[
q_0 v_0 + \cdots + q_n v_n = 0
\]
holds in $N$.  Let $\Sigma$ be the fan made up of the cones generated by all
the proper subsets of $\{v_0,\ldots,v_n\}$.  When $q_i = 1$ for all $i$,
we obtain $X_\Sigma \simeq \P^n$ by Example~\ref{3.Pn}.  And indeed,
$X_\Sigma\simeq\P(q_0,\ldots,q_n) $ in general.  This will be proved in 
Chapter~\ref{homogeneous} using a generalization of the homogeneous coordinates
in $\P^n$.  

Here, we will consider the special case $\P(1,1,2)$, where
$v_0 = -v_1 - 2v_2$.  The fan $\Sigma$ in $N_\R$ is pictured in Figure~5 
in the plane spanned by $v_1,v_2$.
\begin{figure}[ht]
\begin{center}
\includegraphics[height=2.0in]{ch3fig5.ps}
\end{center}
\caption{A fan $\Sigma$ with $X_\Sigma \simeq \P(1,1,2)$} 
\end{figure}
This example has a different property from the others we have
seen so far.  Namely, consider the cone
$\sigma_2 = \Cone(v_0,v_1) = \Cone(-v_1-2v_2,v_1)$.
We have $\sigma_2^\vee = \Cone(-e_2, 2e_1-e_2) \subseteq M$, so 
the situation is similar the case we studied in Example~\ref{1.rnccone2}
in Chapter 1.  Indeed, there is an change of coordinates
defined by a matrix in $\mathrm{GL}(2,\Z)$ that takes $\sigma$ to the cone with
$d = 2$ from that example.  It follows that there is an isomorphism
$U_{\sigma_2} \simeq \V(xz - y^2) \subseteq \C^3$ (Exercise~\ref{3.change}). 
This is the rational normal cone $\widehat{C}_2$, hence has a \emph{singular point} at
the origin.  The toric variety $X_\Sigma$ is singular because of the singular
point in this affine open subset.   We also saw a polytope $P$ such
that the toric variety $X_P \simeq \P(1,1,2)$ in Example~\ref{2.p112fan}
in Chapter~\ref{poly}, and the normal fan $\Sigma_P$ coincides with the 
fan shown above.
\eoe
\end{example}

The following theorem provides a dictionary between properties
of $X_\Sigma$ and properties of $\Sigma$ that generalizes the results
of Theorem~\ref{1.smoothaff} and Example~\ref{1.simplquot} from Chapter~\ref{affine}.  
First, we introduce
some terminology.  The first two items here are 
parallel to those from Definition~\ref{1.regsimp} from Chapter~\ref{affine}.

\begin{definition}
\label{3.fanregsimp}
Let $\Sigma \subseteq N_\R$ be a fan.
\begin{thmenum}
\item We say $\Sigma$ is \emph{\bfseries smooth} (or \emph{\bfseries regular})
if every cone $\sigma$ in $\Sigma$ is smooth (regular).
\item We say $\Sigma$ is \emph{\bfseries simplicial} if every cone $\sigma$
in $\Sigma$ is simplicial.
\item We say $\Sigma$ is \emph{\bfseries complete} if its support
$|\Sigma| = \bigcup_{\sigma\in \Sigma} \sigma$ is all of $N_\R$.
\end{thmenum}
\end{definition}

\begin{theorem}
\label{3.fantoricprops}
Let $X_\Sigma$ be the toric variety defined by a fan $\Sigma \subseteq N_\R$.
\begin{thmenum}
\item $X_\Sigma$ is a smooth variety if and only if the fan $\Sigma$ is smooth.
\item $X_\Sigma$ is an orbifold (that is, $X_\Sigma$ has only finite
quotient singularities) if and only if the fan $\Sigma$ is simplicial.
\item $X_\Sigma$ is compact in its classical topology if and only if 
$\Sigma$ is complete.
\end{thmenum}
\end{theorem}

\begin{proof}  
Part a follows from the corresponding statement for affine 
toric varieties, Theorem~\ref{1.smoothaff}, because smoothness is
a local property (Definition~\ref{3.smoothorb}).
In part b, Example~\ref{1.simplquot} from Chapter 1 gives one 
implication.  The other will be proved in a future chapter.  
A proof of part c will be given in \S 4 of this chapter.
\end{proof}

For example, the blow-up of $\C^2$ at the origin (Example~\ref{3.blowupptfan})
is not compact,
since the support of the cones in the corresponding fan is
not all of $\R^2$.  The Hirzebruch surfaces $\cH_r$ from Example~\ref{3.hirzebruch}
are smooth and compact because every cone in the corresponding fan is smooth, 
and the union of the cones is $N_\R \simeq \R^2$.  The variety $\P(1,1,2)$
from Example~\ref{3.weightedproj} is also compact, but it is not smooth.
It is an orbifold (it has only finite quotient singularities) since the 
corresponding fan is simplicial.

\exercises

\small
\smallskip

\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  from \S 0 for gluing the $V_\sigma$ along
the subvarieties $V_{\sigma_1,\sigma_2}$. 
\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{exercise}
\label{3.Pnex}
Verify that if $\Sigma$ is the fan given in Example~\ref{3.Pn}, 
then $X_\Sigma \simeq \P^n$.
\end{exercise}

\begin{exercise}
\label{3.P1Cstar}
In Example~\ref{3.faninsubspace}, show that the toric variety 
$X_{\Sigma,N'} \simeq \P^1\times \Cstar$.
\end{exercise}

\begin{exercise}
\label{3.blowuporigin}
Let $N \simeq \Z^n$, let $e_1,\ldots,e_n \in N$ be the standard
basis and let $e_0 = e_1 + \cdots + e_n$.  Let $\Sigma$ be the 
set of cones generated by all subsets of 
$\{e_0,\ldots,e_n\}$ \emph{not containing} $\{e_1,\ldots,e_n\}$.
\begin{thmenum}
\item Show that $\Sigma$ is a fan in $N_\R$.
\item Construct the affine open subsets covering the corresponding
toric variety $X_\Sigma$, and give the gluing isomorphisms.
\item Show that $X_\Sigma$ is isomorphic to the blow-up of 
$\C^n$ at the origin described in Exercise~\ref{3.blowups}.
Hint:  The blow-up is the subvariety of $\P^{n-1}\times \C^n$ 
given by $W = \V(x_iy_j - x_jy_i \mid 1 \le i < j \le n)$.  
Cover $W$ by affine open subsets $W_i = W_{x_i}$
and compare those affines with your answer to part b.  
\end{thmenum}  
\end{exercise}

\begin{exercise}
\label{3.change}
In this exercise, you will verify claims made in Example~\ref{3.weightedproj}.
\begin{thmenum}
\item Show that there is a matrix $A \in \mathrm{GL}(2,\Z)$ defining
a change of coordinates that takes the cone in this example
to the cone from Example~\ref{1.rnccone2} from Chapter 1, and find
the mapping takes $\sigma_2^\vee$ to the dual cone.
\item Show that $\Spec(\C[\sS_{\sigma_2}]) \simeq \V(xz - y^2) \subseteq \C^3$.
\end{thmenum}
\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}

\normalsize
\section{The Orbit--Cone Correspondence}

In this section, we will study the orbits for the action 
of $T_N$ on the toric variety $X_\Sigma$.  Our main result
will show that there is a bijective correspondence between the cones 
in $\Sigma$ and the $T_N$-orbits.  The connection comes ultimately
from looking at limit points of \emph{one-parameter subgroups}
of $T_N$ as in Chapter~\ref{affine}, \eqref{1.moncurve}.

\subsection*{\emph{A First Example}}
We introduce the key features by looking first at a concrete example.

\begin{example}
\label{3.P2fanfrom1ps}
Consider $\P^2 \simeq X_\Sigma$ for the fan $\Sigma$ from
Figure~2 of \S 1.  We have $T_N \simeq (\Cstar)^2 \subseteq \P^2$
as the set of points with homogeneous coordinates $(1,s,t)$, $s,t\ne 0$.
For each $n = (a,b) \in N \simeq \Z^2$, we have the corresponding
curve in $\P^2$:
\[
\lambda^n(s) = (1,s^a,s^b).
\]
(We are abusing notation slightly; strictly speaking according
to the definition from Chapter~\ref{affine}, the one-parameter
subgroup $\lambda^n$ is a curve 
in $(\Cstar)^2$, but we view it as a curve in $\P^2$ via the 
inclusion $(\Cstar)^2 \subseteq \P^2$.)  Since $\P^2$ is compact 
(Theorem~\ref{3.fantoricprops}), $\lim_{s\to 0} \lambda^n(s)$ exists 
in $\P^2$ for all $n$.  We start by analyzing how the limit point 
depends on $n$.  It is easy to check that the pattern is as follows.
\begin{eqnarray*}
n = (a,b) & \qquad\qquad & \lim_{s\to 0} \lambda^n(s) \\
&&\\
a = b = 0 &  & (1,1,1) \\
a = 0, b > 0 & & (1,1,0) \\ 
a > 0, b = 0 & & (1,0,1) \\
a = b < 0 & & (0,1,1) \\
a, b > 0 & & (1,0,0) \\
a < 0, b > a & & (0,1,0) \\
b < 0, a > b & & (0,0,1) 
\end{eqnarray*}
For instance, if $a = b = -c < 0$ for some $c > 0$ in $\Z$, by the definition
of the homogeneous coordinates in $\P^2$, $(1,s^{-c},s^{-c}) \sim (s^c,1,1)$
and $c > 0$, so $\lim_{s\to 0} (s^c,1,1) = (0,1,1)$.  You will 
check the remaining cases in Exercise~\ref{3.checkP2}.  

Refering to Figure~2 from \S 1, note the correspondence
between the regions in $N$ described here and the cones in the fan $\Sigma$.  
In each case, we can describe
the set of $n$ giving one of the limit points as $N \cap \mathrm{relint}(\sigma)$, where
$\mathrm{relint}(\sigma)$ denotes the \emph{relative interior} of 
one of the cones $\sigma$ in $\Sigma$.  In other words, we have recovered the structure of the 
fan $\Sigma$ by considering these limits! 

Now we relate this to the $T_N$-orbits in $\P^2$.  By considering the description
$\P^2 \simeq (\C^3 \setminus \{0\})/\Cstar$, you will see in 
Exercise~\ref{3.checkP2} that there are exactly seven $T_N$-orbits
in $\P^2$:
\begin{eqnarray*}
O_1 &=& \{(x_0,x_1,x_2) \mid x_i \ne 0 \text{ for all } i\},\\
O_2 &=& \{(x_0,x_1,x_2) \mid x_2 = 0, \text{ and } x_0,x_1 \ne 0\},\\
O_3 &=& \{(x_0,x_1,x_2) \mid x_1 = 0, \text{ and } x_0,x_2 \ne 0\},\\
O_4 &=& \{(x_0,x_1,x_2) \mid x_0 = 0, \text{ and } x_1,x_2 \ne 0\},\\
O_5 &=& \{(x_0,x_1,x_2) \mid x_1 = x_2 = 0, \text{ and } x_0 \ne 0\} = \{(1,0,0)\},\\
O_6 &=& \{(x_0,x_1,x_2) \mid x_0 = x_2 = 0, \text{ and } x_1 \ne 0\} = \{(0,1,0)\},\\
O_7 &=& \{(x_0,x_1,x_2) \mid x_0 = x_1 = 0, \text{ and } x_2 \ne 0\} = \{(0,0,1)\}.
\end{eqnarray*}
The order these are listed shows the correspondence with the limit points
above.  In each case, note that the limit point lies in the corresponding
orbit.  Hence we obtain a correspondence between cones $\sigma$
and orbits $O$ by 
\[
\sigma \leftrightarrow O \Leftrightarrow \lim_{s\to 0} \lambda^n(s) \in O 
\text{ for all } n \in \mathrm{relint}(\sigma). 
\]
These observations generalize to all toric varieties $X_\Sigma$ and our
goal in the rest of this section is to show this. \eoe
\end{example}

\subsection*{\emph{Points and Semigroup Homomorphisms}}
It will be convenient to use the intrinsic description of the points of 
an affine toric variety $U_\sigma$ given in Proposition~\ref{1.points} from 
Chapter~\ref{affine}.  We recall how this works and make some additional
observations. 
\begin{coxitem}
\item Points of $U_\sigma$ are in bijective correspondence with semigroup
homomorphisms $\gamma : \sS_\sigma \to \C$.
\item For each cone $\sigma$ we have a point of $U_\sigma$ defined by
\begin{equation}
\label{3.distptsg}
m \longmapsto 
\begin{cases}
1 & m \in \sS_\sigma \cap \sigma^\perp\\ 
0 & \text{otherwise}.
\end{cases}
\end{equation}
(This is a well-defined semigroup homomorphism
since $\sigma^\vee \cap \sigma^\perp$ is a face of $\sigma^\vee$. If $m,m'\in \sS_\sigma$ and
$m + m' \in \sS_\sigma \cap \sigma^\perp$, then both $m,m' \in \sS_\sigma \cap \sigma^\perp$.)
We will denote this point by $\gamma_\sigma$ and call it the \emph{distinguished point}
corresponding to $\sigma$.
\item The point $\gamma_\sigma$ is fixed under the $T_N$-action if and only if 
$\dim\sigma = \dim N_\R$ (Corollary~\ref{1.fixptcone} from Chapter~\ref{affine}).
\item If $\tau \subseteq \sigma$ is a face, then $\gamma_\tau \in U_\sigma$.  
(This follows since $\sigma^\perp \subseteq \tau^\perp$.)
\end{coxitem}

\subsection*{\emph{Limits of One-Parameter Subgroups}}
It is not difficult to check that the limit points of one-parameter semigroups 
in Example~\ref{3.P2fanfrom1ps} are exactly the distinguished points for the cones 
in the fan $\Sigma$ for $\P^2$ (Exercise~\ref{3.checkP2}).  Our first general result will
show that the same is true for all affine toric varieties $U_\sigma$.

\begin{proposition}
\label{3.1pslimitpts}
Let $\sigma \subseteq N_\R$ be a strongly convex rational polyhedral cone and let
$n \in N$.  Then
\[
n \in \sigma \Longleftrightarrow \lim_{s\to 0} \lambda^n(s) \text{ exists in } U_\sigma.
\]
Moreover, if $n \in \mathrm{relint}(\sigma)$, then $\lim_{s\to 0} \lambda^n(s) = \gamma_\sigma$.
\end{proposition}

\begin{proof}
Let $m \in \sS_\sigma$.  Then $\chi^m(\lambda^n(s)) = s^{\langle m, n\rangle}$.
Hence $\lim_{s\to 0} \chi^m(\lambda^n(s))$ exists if and only if
$\langle m,n \rangle \ge 0$.  If $n \in \sigma$, then $\langle m,n \rangle \ge 0$
for all $m\in \sS_\sigma = \sigma^\vee\cap M$.  This implies that 
$m \mapsto \lim_{s\to 0} s^{\langle m,n\rangle}$ is a semigroup homomorphism
$\sS_\sigma\to \C$, hence represents a point of $U_\sigma$.  Conversely,
if $\lim_{s\to 0} s^{\langle m,n\rangle}$ exists in $U_\sigma$, then $\langle m,n\rangle \ge 0$
for all $m \in \sigma^\vee$, so $n \in (\sigma^\vee)^\vee = \sigma$.
Finally, if $n \in \mathrm{relint}(\sigma)$, then $\langle m,n\rangle > 0$ for
all $m \in \sS_\sigma \setminus \sigma^\perp$, and
$\langle m,n\rangle = 0$ if $m \in \sS_\sigma\cap\sigma^\perp$.
Thus, the limit point is precisely the distinguished point $\gamma_\sigma$ as above.
\end{proof}

Using this proposition, we can recover the fan $\Sigma$ from $X_\Sigma$ cone by 
cone as in Example~\ref{3.P2fanfrom1ps}.  This is also the key observation needed for the
proof of Corollary~\ref{3.allfromfans} from the previous section.  

When the toric variety $X_\Sigma$ is not compact, note that there will be some
points $n$ for which $\lim_{s\to 0} \lambda^n(s)$
\emph{does not exist}.  For instance, if $X = \C^2$, then the limit point 
$\lim_{s\to 0} \lambda^n(s) = \lim_{s\to 0} (s^a,s^b)$ exists in $X$ 
if and only if $a,b \ge 0$.  There are four different limit points and we
obtain the subfan of $\Sigma$ from Example~\ref{3.P2fanfrom1ps} 
consisting of the cone $\sigma_0$ from
Figure~2 in \S 1, together with its two edges and the origin.  

\subsection*{\emph{The Torus Orbits}}
Now we turn to the $T_N$-orbits in $X_\Sigma$.  
Fix one of the cones $\sigma$ in $\Sigma$.  {}From Exercise~\ref{1.3.1}
of Chapter~\ref{affine}, we know
that $t\in T_N$ acts on the point represented by a semigroup homomorphism
$\gamma : \sS_\sigma \to \C$ to yield the point represented by the 
semigroup homomorphism
\begin{equation}
\label{3.actonpt}
t\cdot \gamma : m \longmapsto \chi^m(t) \gamma(m).
\end{equation}
{}From this description, it is easy to identify a certain subset of $U_\sigma$
that is invariant under the $T_N$-action.

\begin{lemma}
\label{3.invar}
Let $\sigma$ be a strongly convex rational polyhedral cone, and 
define $O(\sigma)$ to be the following set of semigroup homomorphisms
representing points in a subset of $U_\sigma$:
\begin{eqnarray*}
O(\sigma) &=& \{ \gamma : \sS_\sigma \to \C \mid 
\gamma(m) \ne 0\, \text{  for all } m \in \sS_\sigma \cap \sigma^\perp, \\
          & &\qquad\qquad\qquad \gamma(m) = 0 \text{  for all } m \in \sS_\sigma \setminus \sigma^\perp \}.
\end{eqnarray*}
Then $O(\sigma)$ is nonempty and invariant under the $T_N$-action on $U_\sigma$.
\end{lemma}

\begin{proof}
The distinguished point $\gamma_\sigma$ lies in $O(\sigma)$, so $O(\sigma)$ is not
empty.  Invariance under $T_N$ follows immediately from the definition and
\eqref{3.actonpt}.
\end{proof}

In fact, we will show next that $O(\sigma)$ is a $T_N$-orbit in $U_\sigma$.  To see
this we need to make several identifications.

First, notice that $\sigma^\perp$ is the
largest vector subspace of $M_\R$ contained in $\sigma^\vee$.
Hence $\sS_\sigma \cap \sigma^\perp$ is actually a \emph{subgroup} of $\sS_\sigma$.  
If $\gamma(m) \ne 0$ for all $m \in \sS_\sigma\cap\sigma^\perp$,
then restricting $\gamma$ to $m\in \sS_\sigma\cap\sigma^\perp = M\cap \sigma^\perp$ yields a
\emph{group homomorphism} $\widehat{\gamma} : M \cap\sigma^\perp \to \Cstar$ 
(see Exercise~\ref{3.quottorus}).
Conversely, given any group homomorphism $\widehat{\gamma}: M\cap\sigma^\perp \to \Cstar$, 
we obtain a semigroup homomorphism $\gamma \in O(\sigma)$ by defining
\[
\gamma(m) = \begin{cases}
            \widehat{\gamma}(m) & \text{ if } m \in M\cap\sigma^\perp\\
                  0 & \text{ otherwise }.
            \end{cases}
\]
The argument sketched here can be completed to show that $O(\sigma)$
has the structure of a group and that there
is an isomorphism of groups 
\begin{equation}
\label{3.orbtorus}
O(\sigma) \simeq \mathrm{Hom}_\Z(M\cap \sigma^\perp, \Cstar)
\end{equation}
(you will check the details in Exercise~\ref{3.quottorus}).

Now recall Exercise~\ref{1.1.10} from Chapter~\ref{affine}, in which
you showed that if $T_N$ is a torus with character lattice $M$, then
there is an isomorphism of groups
\[
T_N \simeq \mathrm{Hom}_\Z(M, \Cstar).
\]
The right side of \eqref{3.orbtorus} is analogous to the 
right side here, since $M\cap \sigma^\perp$ is a lattice in the
subspace $\sigma^\perp \subseteq M_\R$.  What plays the role of 
$N$, though?  We claim that the answer is given as in the following
Lemma.

\begin{lemma}
\label{3.duallattices}
Let $\sigma$ be the cone in $N_\R$ in the above discussion.  Let
$N_\sigma$ be the sublattice of $N$ spanned by the points in $\sigma\cap N$,
and let $N(\sigma) = N/N_\sigma$.  There is a dual pairing 
\[
\langle\ ,\ \rangle : M\cap \sigma^\perp \times N(\sigma) \to \Z,
\]
induced by the dual pairing 
$\langle\ ,\ \rangle : M \times N \to \Z$. \qed
\end{lemma}

You will prove this in Exercise~\ref{3.duals}.
The same reasoning as in Exercise~\ref{1.1.10} from Chapter~\ref{affine}
now shows that we have identified $O(\sigma)$ as
\begin{equation}
\label{3.orbastorus}
O(\sigma) \simeq \mathrm{Hom}_\Z(M\cap\sigma^\perp,\Cstar) \simeq T_{N(\sigma)}.
\end{equation}
The exact sequence 
\begin{equation}
\label{3.coneexseq}
0 \longrightarrow N_\sigma \longrightarrow N \longrightarrow N(\sigma) \longrightarrow 0
\end{equation}
implies that $T_{N(\sigma)} \simeq N(\sigma)\otimes_\Z \Cstar$ is a quotient of 
$T_N\simeq N\otimes_\Z \Cstar$.  The action of $T_N$ 
on this quotient torus is transitive, so $O(\sigma)$ is a $T_N$-orbit
in $U_\sigma$.

\subsection*{\emph{The Orbit-Cone Correspondence}}
Our next theorem is the major result of this section.

\begin{theorem}[Orbit-Cone Correspondence]
\label{3.occorresp}
Let $X_\Sigma$ be the toric variety associated to the fan $\Sigma$ in $N_\R$.
\begin{thmenum}
\item There is a bijective correspondence
\begin{eqnarray*}
\{ \text{\rm cones } \sigma \text{\rm\ in } \Sigma\}
&\longleftrightarrow& \{T_N-\text{\rm orbits in } X_\Sigma \}\\
\sigma &\longleftrightarrow& O(\sigma) \simeq \mathrm{Hom}_\Z(M\cap\sigma^\perp, \Cstar).
\end{eqnarray*}
\item Let $k = \dim_\R N_\R$.  
For each cone $\sigma$, $\dim_\C O(\sigma) = k - \dim_\R \sigma$.
\item $O(\sigma)$ is the unique closed $T_N$-orbit in the affine open
set $U_\sigma$, and 
\begin{equation}
\label{3.orbitscover}
U_\sigma = \bigcup_{\{\tau \mid \tau \text{\rm\ is a face of } \sigma\}} O(\tau).
\end{equation}
\item $\tau$ is a face of $\sigma$ if and only if $O(\sigma) \subseteq \overline{O(\tau)}$, 
and 
\begin{equation}
\label{3.orbitclosure}
\overline{O(\tau)} = \bigcup_{\{\sigma\mid\tau \text{\rm\ is a face of } \sigma\}} O(\sigma).
\end{equation}
\end{thmenum}
\end{theorem}

For instance, in Example~\ref{3.P2fanfrom1ps}, there are three types
of cones and torus orbits in the variety $\P^2$.
\begin{coxitem}
\item The cone $\rho = \{(0,0,0)\}$ corresponds to the orbit $O(\rho) = T_N \subseteq \P^2$, 
which satisfies $\dim_\C O(\rho) = 2 = 2 -\dim_\R \rho$.  This $\rho$ 
is a face of all the other cones in $\Sigma$, and hence all 
the other orbits are contained in the closure of this
one by part d.  According to part c, since there are no 
cones properly contained in $\rho$, $U_\rho = O(\rho) \simeq (\Cstar)^2$.  
\item The three one-dimensional cones $\tau$ give the torus orbits of dimension 1.
Each of these is isomorphic to $\Cstar$.  The closure of one
of these orbits is one of the coordinate axes $\V(x_i) \subseteq \P^2$,
a copy of $\P^1$.
Note that each $\tau$ is contained in two maximal cones.  
\item The three maximal cones $\sigma_i$ in the fan $\Sigma$ 
correspond to the three fixed points $(1,0,0), (0,1,0), (0,0,1)$ of 
the torus action on $\P^2$.   There are two of these in the closure
of each of the 1-dimensional torus orbits.
\end{coxitem}

\begin{proof}
Proposition~\ref{3.1pslimitpts} shows that the statement of part a
is true for the affine toric variety $U_\sigma$ and the faces $\tau\subseteq\sigma$. 
We have
\[
\tau \leftrightarrow O = O(\tau) \Leftrightarrow \lim_{s\to 0} \lambda^n(s) = \gamma_\tau \in O 
\text{ for all } n \in \mathrm{relint}(\tau). 
\]
The variety $X_\Sigma$ is obtained by gluing several of these affine opens, and part 
a follows in general from the fact that the gluing isomorphisms are 
$T_N$-equivariant.  If $\tau = \sigma_1\cap\sigma_2$, then the orbits
$O(\rho) \subseteq U_{\sigma_1}$ and $O(\rho)\subseteq U_{\sigma_2}$ 
for all faces $\rho$ of $\tau$ are identified in the gluing.  Part b follows from 
\eqref{3.orbastorus} and the following comments.

We will prove parts c and d together.  We start with \eqref{3.orbitscover}
from part c.
If $\gamma$ is a semigroup homomorphism $\gamma : \sS_\sigma \to \C$ representing
a point $p \in U_\sigma$, then in Exercise~\ref{3.nonzeroface}, you 
will show that $\{m \in \sS_\sigma \mid \gamma(m) \ne 0\} = \rho \cap M$
for some face $\rho$ of $\sigma^\vee$.  By Proposition~\ref{1.dualfaceprops} from
Chapter~\ref{affine}, $\rho = \sigma^\vee \cap \tau^\perp$ for some 
face $\tau$ of $\sigma$.  Hence 
\[
\{m \in \sS_\sigma \mid \gamma(m) \ne 0\} = \sS_\sigma \cap \tau^\perp,
\]
and the set of all $\gamma$ which are nonzero precisely on this set is just
the subset $O(\tau)$.  This shows
\[
U_\sigma \subseteq \bigcup_{\{\tau\mid \tau \text{\rm\ is a face of } \sigma\}} O(\tau).
\]
The opposite inclusion is clear, so \eqref{3.orbitscover}
follows.  Since the $O(\tau)$ are orbits for a group action,
note that this is a disjoint union.  Moreover, since $U_\sigma$
is an affine toric variety, it is invariant under the $T_N$-action
and the $O(\tau)$ for faces $\tau\subseteq\sigma$ are the only orbits 
that intersect $U_\sigma$.

Next, we turn to part d.  If $O(\sigma) \subseteq \overline{O(\tau)}$,
then $O(\tau) \subseteq U_\sigma$, hence $\tau$ is a face of $\sigma$
by \eqref{3.orbitscover}.  

Conversely, if $\tau$ is a face
of $\sigma$, then $\sigma^\vee \cap \tau^\perp$ is a face of $\sigma^\vee$.
We want to show that $O(\sigma) \subseteq \overline{O(\tau)}$.  To prove
this, it suffices to show that $\overline{O(\tau)} \cap O(\sigma) \ne \emptyset$
(Exercise~\ref{3.orbfact}).  We will do this by using limits of one-parameter
semigroups as in Proposition~\ref{3.1pslimitpts}.

Let $\gamma_\tau$ be the semigroup homomorphism corresponding to 
the distinguished point of $U_\tau$, so $\gamma_\tau(m) = 1$ if $m \in M\cap\tau^\perp$,
and $0$ otherwise.
Let $n\in\mathrm{relint}(\sigma)$, and for each $s\in \Cstar$ consider
$\gamma(s) = \lambda^n(s)\cdot \gamma_\tau$.  This is a semigroup homomorphism
which maps
\[
m\longmapsto \chi^m(\lambda^n(s))\gamma_\tau(m) = s^{\langle m,n\rangle} \gamma_\tau(m).
\]
For all $s\in \Cstar$, $\gamma(s) \in O(\tau)$ since the orbit of $\gamma_\tau$
is $O(\tau)$.  Now let $s \to 0$.  Since $n\in \mathrm{relint}(\sigma)$, 
$\langle m, n\rangle > 0$ if $m \in \sigma^\vee \setminus \sigma^\perp$, and
$ = 0$ if $m \in \sigma^\perp$.
It follows that $\gamma(0) = \lim_{s\to 0} \gamma(s)$ exists as a point in $U_\sigma$
by Proposition~\ref{3.1pslimitpts}, and represents a point in $O(\sigma)$. 
But it is also in the closure of $O(\tau)$ by construction, so 
$O(\sigma) \cap \overline{O(\tau)} \ne \emptyset$.  This establishes the first part of d,
and \eqref{3.orbitclosure} follows from that.

Finally, the first part of c is a consequence of the rest (Exercise~\ref{3.finishocc}).
\end{proof}

In the example of $\P^2$, 
the orbit closures $\overline{O(\tau)}$ also have the structure of toric varieties.
The same is true in general.
The torus that acts is the quotient torus $T_{N(\tau)}$ from \eqref{3.orbtorus}.
The corresponding fan can be described as follows.  Consider the set of 
cones $\sigma$ in $\Sigma$ containing $\tau$ as a face.  For each such
$\sigma$, let $\overline{\sigma}$ be the image cone in $N(\tau)_\R$ under
the quotient map
\[
N_\R \longrightarrow N(\tau)_\R \longrightarrow 0
\]
obtained from \eqref{3.coneexseq}.  Then
\begin{equation}
\label{3.star}
\mathrm{Star}(\tau) = \{\overline{\sigma} \subseteq N(\tau)_\R \mid \tau \text{ is a face of } \sigma\}
\end{equation}
is a fan in $N(\tau)_\R$ (Exercise~\ref{3.starcone}).  

\begin{corollary}
\label{3.starfan}
For each cone $\tau$ in $\Sigma$, the orbit closure $\overline{O(\tau)}$
is isomorphic to the toric variety $X_{\mathrm{Star}(\tau)}$.  
\end{corollary} 

\begin{proof}  This follows from parts a and d of Theorem~\ref{3.occorresp}
(Exercise~\ref{3.starcone}).
\end{proof}

\subsection*{\emph{The Variety $\V(xy - zw)$}}
We next turn to another example.  

\begin{example}
\label{3.xyzwagain}
Consider the affine toric variety $X = \V(xy-zw)$ studied in a number of examples
from Chapter~\ref{affine}.  For instance, in Example~\ref{1.xyzw2} from 
that Chapter, we showed that $X$ is the normal toric variety
corresponding to some cone $\sigma$.  The dual cone is
\begin{equation}
\label{3.Segredualcone}
\sigma^\vee = \Cone(e_1,e_2,e_3,e_1+e_2-e_3),
\end{equation}
and $X = \Spec(\C[M \cap \sigma^\vee])$.

Let us see how the results of this section apply to this 
example.  In Example~\ref{1.xyzw2} from Chapter~\ref{affine}, we introduced
the torus $T = (\Cstar)^3$ included in $X$ as the image of 
\begin{equation}
\label{3.T3}
(t_1,t_2,t_3) \mapsto (t_1,t_2,t_3,t_1t_2t_3^{-1}).
\end{equation}
Given $n = (a,b,c) \in N = \Z^3$, we have the one-parameter subgroup
\begin{equation}
\label{3.1ps}
\lambda^n(s) = (s^a,s^b,s^c,s^{a+b-c})
\end{equation}
contained in $X$, and we proceed to examine limit points.  Clearly, the limit
point is in $X$ if and only if $a,b,c\ge 0$ and $a + b \ge c$.  These conditions
determine the cone $\sigma \subseteq N_\R$ given by 
\begin{equation}
\label{3.Segrecone}
\sigma = \Cone(e_1,e_2,e_1+e_3,e_2+e_3).
\end{equation}
%We have the following possible limit points:
%\begin{eqnarray*}
%n = (a,b,c) & \qquad\qquad & \lim_{s\to 0} \lambda^n(s) \\
%&&\\
%a = b = c = 0 &  & (1,1,1,1), \\
%a > 0, b = c = 0 & & (0,1,1,0), \\ 
%b > 0, a = c = 0 & & (1,0,1,0), \\
%a = c > 0, b = 0 & & (0,1,0,1), \\
%b = c > 0, a = 0 & & (1,0,0,1), \\
%a,b > 0, c = 0 & & (0,0,1,0), \\
%a > 0, 0 < c < a, b = 0 & & (0,1,0,0), \\
%a,b > 0, c = a+b, & & (0,0,0,1), \\
%a = 0, 0 < c < b, b > 0 & & (1,0,0,0), \\
%a,b,c > 0, a + b > c & & (0,0,0,0).
%\end{eqnarray*}
%As we expect from the Orbit-Cone Correspondence,
%this clearly reflects the structure of the strongly convex rational
%cone $\sigma$, which has its vertex at the origin,
%four edges, four two-dimensional faces, and whose interior is open
%in $\N_\R$.

It is easy to check that this cone and the cone from \eqref{3.Segredualcone}
are in fact dual to each other (Exercise~\ref{3.Segrecones}).
\eoe
\end{example} 

\subsection*{\emph{Final Comments}}
The technique of using limit points of one-parameter subgroups to study
a group action is also a major tool in Geometric Invariant Theory
as in \cite{mfk}, where the main problem is to construct
varieties (or possibly more general objects) representing orbit spaces for 
the actions of algebraic groups on varieties.  We will see how some of the 
ideas from that branch of algebraic geometry apply in the study of toric
toric varieties in Chapter~\ref{homogeneous} and Chapter~\ref{git}.

\exercises

\small

\begin{exercise}
\label{3.checkP2}
In this exercise, you will verify the claims made in Example~\ref{3.P2fanfrom1ps}
and the following discussion.
\begin{thmenum}
\item Show that the remaining limits of one-parameter subgroups $\P^2$
are as claimed in the example.
\item Show that the $(\Cstar)^2$ orbits in $\P^2$ are as claimed in the example.
\item Show that the limit point equals the distinguished point $p_\sigma$
of the corresponding cone in each case.  
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{3.quottorus}
In this exercise, you will supply some of the details
that let us identify $O(\sigma)$ from Lemma~\ref{3.invar}
as a $T_N$-orbit in $U_\sigma$.
\begin{thmenum}
\item Show that if $\gamma \in O(\sigma)$, 
then $\widehat{\gamma} : M\cap\sigma^\perp \to \Cstar$ is 
a group homomorphism.
\item Deduce that $O(\sigma)$ has the structure of a group.
\item Verify carefully that we have an isomorphism of groups
$O(\sigma) \simeq \mathrm{Hom}_\Z(M\cap\sigma^\perp,\Cstar)$.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{3.duals}
Prove Lemma~\ref{3.duallattices}
\end{exercise}

\begin{exercise}
\label{3.nonzeroface}
Show that if $\gamma: \sS_\sigma \to \C$ is a 
semigroup homomorphism, then 
$\{m \in \sS_\sigma \mid \gamma(m) \ne 0\} = \rho \cap M$
for some face $\rho$ of $\sigma^\vee$.
\end{exercise}

\begin{exercise}
\label{3.orbfact}
Show that if $\tau,\sigma$ are two cones and 
$\overline{O(\tau)} \cap O(\sigma) \ne \emptyset$, 
then $O(\sigma) \subseteq \overline{O(\tau)}$.  
Hint: Show $\overline{O(\tau)}$ is invariant under
the action of $T_N$.  
\end{exercise}

\begin{exercise}
\label{3.finishocc}
Show that \eqref{3.orbitscover} and \eqref{3.orbitclosure}
imply the first part of c in Theorem~\ref{3.occorresp}.
\end{exercise}

\begin{exercise}
\label{3.openface}
In Proposition~\ref{1.facesg} of Chapter 1, we saw that if $\tau$ is a face
of the strongly convex rational polyhedral cone $\sigma$ in $N_\R$
then $U_\tau = \Spec(\C[\sS_\tau])$ is an affine open subset
of $U_\sigma = \Spec(\C[\sS_\sigma])$.  In this exercise, we
will show that the converse is also true.  Namely we will see that if 
$\tau \subseteq \sigma$ and the induced map of affine
toric varieties $\varphi : U_\tau \to U_\sigma$ 
is an open immersion, then $\tau$ is a face of $\sigma$.  
\begin{thmenum}
\item Let $n,n' \in N\cap \sigma$, and assume $n + n' \in \tau$.
Show that $\lim_{s\to 0} \lambda^n(s) \cdot \lim_{s\to 0} \lambda^{n'}(s) \in U_\tau$.
\item Show that $\lim_{s\to 0} \lambda^n(s)$ and $\lim_{s\to 0} \lambda^{n'}(s)$
are each in $U_\tau$.  Hint: use the description of points as semigroup homomorphisms.  
\item Deduce that $n,n'\in \tau$, so $\tau$ is a face of $\sigma$.
\end{thmenum}
\end{exercise}


\begin{exercise}
\label{3.Cor17}
In this exercise, you will
use Proposition~\ref{3.1pslimitpts} and Theorem~\ref{3.occorresp} to
deduce Corollary~\ref{3.allfromfans} from Theorem~\ref{3.sumihiro}.
\begin{thmenum}
\item By Theorem~\ref{3.sumihiro}, and the results of 
Chapter~\ref{affine}, a separated toric variety has 
an open cover consisting of affine toric varieties $U_i = U_{\sigma_i}$
for some collection of cones $\sigma_i$.  Show that for all
$i,j$, $U_i \cap U_j$ is also affine.  Hint: Use the fact that $X$ is separated.
\item Show that $U_i \cap U_j$ is the affine toric variety corresponding
to the cone $\tau = \sigma_i \cap \sigma_j$.  Hint: Exercise~\ref{3.1pslimitpts}
will be useful.
\item If $\tau = \sigma_i \cap \sigma_j$, then show that
$\tau$ is a face of both $\sigma_i$ and $\sigma_j$.  Hint: See Exercise~\ref{3.openface}
above.
\item Deduce that $X \simeq X_\Sigma$ for the fan consisting of the $\sigma_i$
and all their faces.  
\end{thmenum}
\end{exercise}
\begin{exercise}  

\label{3.starcone}
Let $\tau$ be a cone in a fan $\Sigma$, and let
$\mathrm{Star}(\tau)$ be as defined in \eqref{3.star}.
\begin{thmenum}
\item Show that $\mathrm{Star}(\tau)$ is a fan in $N(\tau)_\R$.
\item Prove Corollary~\ref{3.starfan}.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{3.Segrecones}
In this exercise, you will check the claims made in Example~\ref{3.xyzwagain}.
\begin{thmenum}
\item Show that the cones in \eqref{3.Segrecone} and \eqref{3.Segredualcone}
are dual.
\item Identify the limits of one-parameter semigroups in this 
example, and describe the orbit-cone correspondence in this case.
\item Show that the matrix 
\[
A = \begin{pmatrix} 1 & 1 & -1\\ 1 & 0 & 0\\ -1 & 0 & 1\end{pmatrix}
\]
defines an automorphism of $N \simeq \Z^3$ and the corresponding
linear map on $N_\R$ maps the cone $\sigma^\vee$ to $\sigma$.
\item Deduce that the two affine toric varieties $U_\sigma$ and $U_{\sigma^\vee}$
are isomorphic.  Hint:  Use Proposition~\ref{1.3.conemorphism}
from Chapter 1. 
\end{thmenum}
\end{exercise}






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

\normalsize

\section{Equivariant Maps of Toric Varieties}

Recall from \S 0 of this chapter that if $X$ and $Y$
are varieties with affine open covers 
$X = \bigcup_\alpha U_\alpha$ and $Y = \bigcup_\beta U'_\beta$,
then a  morphism of varieties $\varphi : X \to Y$
is a Zariski-continuous mapping such that the 
restrictions
\[
\varphi\res{U_\alpha\cap \varphi^{-1}(U'_\beta)} : U_\alpha \cap \varphi^{-1}(U'_\beta)
\longrightarrow U'_\beta 
\]
are morphisms of affine varieties for all $\alpha,\beta$. 

When $X$ and $Y$ are normal toric varieties, the results on 
mappings of affine toric varieties from Proposition~\ref{1.toricmapprop}
and Proposition~\ref{1.3.conemorphism} from Chapter~\ref{affine}
yield a class of morphisms whose construction comes directly
from the combinatorics of the associated fans.  Our goal in this section
is to study these special toric morphisms.  We start by introducing some terminology.  

\begin{definition}
\label{3.compatiblemap}
Let $N,N'$ be two lattices, let $\Sigma$ be a fan in $N_\R$, and let
$\Sigma'$ be a fan in $N'_\R$.  A $\Z$-linear mapping
$\overline{\varphi} : N \to N'$ is \emph{\bfseries compatible} with the fans 
$\Sigma$ and $\Sigma'$ if for every cone $\sigma$ in $\Sigma$, there exists a cone 
$\sigma'$ in $\Sigma'$ such that $\overline{\varphi}_\R(\sigma) \subseteq \sigma'$.  
\end{definition}

\begin{example}
\label{3.HP1Ex1}
Let $N \simeq \Z^2$ with basis $\{e_1,e_2\}$
and $\Sigma$ be the fan from Figure~4 in \S~1, 
so from Example~\ref{3.hirzebruch},
$X_\Sigma$ is the Hirzebruch surface $\cH_r$.  Let $N' \simeq \Z$
with basis $\{f\}$ and consider the fan $\Sigma'$
\[
\begin{picture}(220,25)
\thicklines
\put(10,15){\line(1,0){200}}
\put(110,15){\circle*{4}}
\put(108,4){$0$}
\put(60,5){$\sigma_1$}
\put(160,5){$\sigma_0$}
\end{picture}
\]
where $\sigma_1 = \Cone(-f)$, and $\sigma_0 = \Cone(f)$.
$X_{\Sigma'} \simeq \P^1$ (see Example~\ref{2.P1P1} in 
Chapter~\ref{poly}).  The mapping 
\[
\overline{\varphi} : N \to N', \quad\quad  ae_1 + be_2 \mapsto af
\]
is compatible with the fans $\Sigma$ and 
$\Sigma'$ since each cone in $\Sigma$ maps onto one of the cones in $\Sigma'$.
If $r \ne 0$, on the other hand, the mapping 
\[
\overline{\psi} : N \to N', \quad\quad  ae_1+be_2 \mapsto bf
\]
is not compatible with these fans since the cone $\sigma_3$ 
in $\Sigma$ does not map into just one of the cones of $\Sigma'$.  
\eoe
\end{example}

\subsection*{\emph{Toric morphisms}}
It will be convenient in the following to take the result of 
Proposition~\ref{1.toricmapprop}, part a, from Chapter~\ref{affine} as the 
\emph{definition} of a toric morphism in this context.  
The discussion from Chapter~\ref{affine}
shows that this is consistent with what we did before for affine 
toric varieties.

\begin{definition}
\label{3.toricmorph}
Let $X_\Sigma$, $X_{\Sigma'}$ be normal toric varieties, with $\Sigma$ a fan
in $N$ and $\Sigma'$ a fan in $N'$. 
A morphism $\varphi : X_\Sigma \to X_{\Sigma'}$ is \emph{\bfseries toric}
if $\varphi$ maps the torus $T_N \subseteq X_\Sigma$ into 
$T_{N'} \subseteq X_{\Sigma'}$ and $\varphi\res{T_N}$
is a group homomorphism.
\end{definition} 

The proof of part b of Proposition~\ref{1.toricmapprop} from Chapter 1
generalizes immediately to show that any toric morphism 
$\varphi : X_\Sigma \to X_{\Sigma'}$ is an \emph{equivariant mapping}
for the $T_N$- and $T_{N'}$-actions.  That is, we have a commutative diagram
\begin{equation}
\label{3.equivar}
\SelectTips {cm}{}
\xymatrix{
T_N\! \times X_\Sigma \ar[r]^(.6){\Phi}
 \ar[d]_{\varphi\smallres{T_N}\!\times
  \varphi} &  X_\Sigma \ar[d]^{\varphi}\\
T_{N'}\! \times X_{\Sigma'} \ar[r]^(.6){\Phi'}  &  X_{\Sigma'}
}
\end{equation}
where $\Phi$ and $\Phi'$ give the torus actions.

Our first result in this section shows that toric morphisms
$\varphi : X_\Sigma \to X_{\Sigma'}$ are in bijective correspondence
with $\Z$-linear mappings
$\overline{\varphi} : N \to N'$ that are compatible with the 
fans $\Sigma$ and $\Sigma'$.

\begin{theorem}
\label{3.mapcorresp}
Let $N,N'$ be lattices, and let $\Sigma$ in $N$ and $\Sigma'$
in $N'$ be fans.  
\begin{thmenum}
\item If $\overline{\varphi} : N \to N'$ is a $\Z$-linear
map that is compatible with $\Sigma$ and $\Sigma'$, then 
there is a corresponding toric morphism 
$\varphi : X_\Sigma \to X_{\Sigma'}$ with the property
that $\varphi\res{T_N}$ is the map 
\[
\overline{\varphi}\otimes 1 : N\otimes_\Z \C \to N' \otimes_\Z \C.
\]
\item  Conversely, if $\varphi : X_\Sigma \to X_{\Sigma'}$ is a 
toric morphism, then $\varphi$ induces a $\Z$-linear map 
$\overline{\varphi} : N \to N'$
that is compatible with the fans $\Sigma$ and $\Sigma'$.
\end{thmenum}
\end{theorem}

\begin{proof}
To prove part a, let $\sigma$ be a cone in $\Sigma$.  
Because $\varphi$ is compatible with the fans $\Sigma$ and $\Sigma'$, 
there is some cone $\sigma'$ in $\Sigma'$ such that 
$\overline{\varphi}_\R(\sigma) \subseteq \sigma'$.  Then
Proposition~\ref{1.3.conemorphism} from Chapter~\ref{affine}
shows that $\overline{\varphi}$ induces a toric morphism 
$\varphi_\sigma : U_\sigma \to U_{\sigma'}$.  
Using the general criterion for gluing morphisms 
from Exercise~\ref{3.gluemaps}, you will show in 
Exercise~\ref{3.toricgluemaps} that the $\varphi_\sigma$ glue
together to give a morphism $\varphi : X_\Sigma \to X_{\Sigma'}$.
Moreover, $\varphi$ is toric because  
the $\Z$-linear map $\overline{\varphi} : N \to N'$
induces the group homomorphism
$\varphi : T_N  \to T_{N'}$
on $T_N \subseteq U_\sigma$ for all $\sigma$ by the comments
after Proposition~\ref{1.toricmapprop} in Chapter~\ref{affine}.

For part b, we show first that the toric morphism
$\varphi$ induces a $\Z$-linear map 
$\overline{\varphi} : N \to N'$.  This follows since $\varphi\res{T_N}$
is a group homomorphism.  The lattices $M,M'$ are the lattices of 
characters of the two tori.  For $\chi' \in M'$, 
the mapping $\chi' \mapsto \chi'\circ \varphi\res{T_N}$ defines
a group homomorphism $M' \to M$.  Hence the dual of this map 
is the $\overline{\varphi} : N \to N'$ we want. 

It remains to show that  $\overline{\varphi}$ is 
compatible with the lattices $\Sigma$ and $\Sigma'$. 
Because of the equivariance \eqref{3.equivar},
each $T_N$-orbit $O \subseteq X_{\Sigma}$ is mapped into a $T_{N'}$-orbit
$O' \subseteq X_{\Sigma'}$.   By the Orbit-Cone Correspondence 
(Theorem~\ref{3.occorresp}, part a), each $T_N$-orbit is $O = O(\sigma)$ for
some cone $\sigma$ in $\Sigma$, and similarly each $T_{N'}$-orbit
is $O' = O(\sigma')$ for some cone $\sigma'$ in $\Sigma'$.  
Furthermore, if $\tau \subseteq \sigma$ is a face, then by the same
reasoning, there is some cone $\tau'$ in $\Sigma'$ such that
$\varphi(O(\tau)) \subseteq O(\tau')$.  

We claim that in this situation $\tau'$ must be a face of $\sigma'$.  
This follows since $O(\sigma) \subseteq \overline{O(\tau)}$ by part 
d of Theorem~\ref{3.occorresp}.  Since $\varphi$ is continuous in 
the classical topology, $\varphi\left(\overline{O(\tau)}\right) \subseteq \overline{O(\tau')}$.
But the only orbits contained in the closure of $O(\tau')$
are the orbits corresponding to cones which have $\tau'$ as 
a face.  So $\tau'$ is a face of $\sigma'$.   It follows 
from part c of Theorem~\ref{3.occorresp} that $\varphi$ 
also maps the affine open subset $U_\sigma\subseteq X_\Sigma$ 
into $U_{\sigma'} \subseteq X_{\Sigma'}$:
\begin{equation}
\label{3.preserveaffine}
\varphi(U_\sigma) \subseteq U_{\sigma'}.
\end{equation}

Now we use limit points of one-parameter semigroups once again
to show that $\overline{\varphi}_\R(\sigma) \subseteq \sigma'$.
By Proposition~\ref{3.1pslimitpts}, if $n \in N\cap \sigma$,
then $\lim_{s\to 0} \lambda^n(s)$ exists in $U_\sigma$. 
By \eqref{3.preserveaffine}, $\varphi(\lim_{s\to 0} \lambda^n(s))$ is 
a point of $U_{\sigma'}$.  But by continuity of $\varphi$,
\begin{eqnarray*}
\varphi(\lim_{s\to 0} \lambda^n(s)) &=& \lim_{s\to 0} \varphi(\lambda^n(s))\\
&=& \lim_{s\to 0} \lambda^{\overline{\varphi}(n)}(s).
\end{eqnarray*}
Therefore, by Proposition~\ref{3.1pslimitpts} again, $\overline{\varphi}(n) \in \sigma'$.
It follows easily that $\overline{\varphi}_\R(\sigma) \subseteq \sigma'$
since the cones are rational.  Hence $\overline{\varphi}$ is compatible
with the fans $\Sigma$ and $\Sigma'$.
\end{proof}

\subsection*{\emph{First examples of toric morphisms}}
Here are some first examples of toric morphisms defined by
mappings compatible with the corresponding fans.

\begin{example}
\label{3.HP1Ex2}
Let $N \simeq \Z^2$, $N' \simeq \Z$, and let 
\[
\overline{\varphi} : N  \to N', \qquad \qquad
            ae_1 + be_2 \mapsto af,
\]
(the first mapping in Example~\ref{3.HP1Ex1}).
We saw that $\overline{\varphi}$ is compatible with the fans
$\Sigma$ definining the Hirzebruch surface and $\Sigma'$
defining $\P^1$.  Theorem~\ref{3.mapcorresp} implies
there is a corresponding toric morphism $\varphi: \cH_r \to \P^1$.
We will see later in this section that this mapping gives
$\cH_r$ the structure of a $\P^1$-bundle over $\P^1$.
\eoe
\end{example} 

\begin{example}
\label{3.pospowermap}
Let $\Sigma$ be any fan in $N \simeq \Z^n$.  If $k\in \Z_{\ge 0}$,
then the multiplication map 
\[
\overline{\varphi} : N \to N, \qquad\qquad n \mapsto k\cdot n
\]
is compatible with $\Sigma$.  Theorem~\ref{3.mapcorresp}
shows that there is a corresponding toric morphism
$\varphi : X_\Sigma \to X_\Sigma$, whose restriction to 
$T_N \subseteq X_\Sigma$ is the group endomorphism
\[
\varphi\res{T_N}(t_1,\ldots,t_n) = (t_1^k,\ldots,t_n^k).
\]
For instance, if $\Sigma$ is the fan in $N\simeq \Z^2$ from 
Figure~2 and $k = 2$, then we obtain the morphism $\varphi : \P^2 \to \P^2$
defined in homogeneous coordinates by $\varphi(x_0,x_1,x_2) = (x_0^2,x_1^2,x_2^2)$.
\eoe
\end{example}

\subsection*{\emph{Change of lattice}}
Toric morphisms may also induced by the
inclusion of one lattice in another.  Let $N'$ be a
lattice and let  $N \subseteq N'$ be a sublattice.
If $\Sigma'$ is a fan in whose cones are generated
by elements of $N$, then we can view $\Sigma'$ as a
fan either in $N_\R$ or in $N'_\R$, and the 
inclusion map $\overline{\varphi} : N \to N'$
is compatible with the fan $\Sigma'$ in the domain and codomain.   
As in Chapter~\ref{affine}, we obtain two 
different toric varieties $X_{\Sigma',N}$ and
$X_{\Sigma',N'}$ depending on which lattice we consider, 
and the inclusion induces a toric morphism
\[
\varphi : X_{\Sigma, N} \to X_{\Sigma, N'}.
\]

First let us consider the case where $N$ is spanned by 
a subset of a basis for $N'$.  We saw an instance of this in 
Example~\ref{3.faninsubspace} from \S 1, and the general
situation is very similar.  

\begin{proposition}
\label{3.splitsublattice}
Let $N \subseteq N'$ be a sublattice spanned by a subset
of a basis for $N'$.  Let $\Sigma$ be a fan in $N_\R$.
Then we have an isomorphism 
\[
\varphi : X_{\Sigma,N'} \simeq X_{\Sigma,N} \times T_{N'/N},
\]
where if $\dim N_\R = n$ and $\dim (N')_\R = k$, then 
$T_{N'/N} \simeq (\Cstar)^{n-k}$.
\end{proposition}

\begin{proof}
This was proved for a fan consisting of a single 
cone $\sigma$ and all of its faces in the course of the proof 
of Theorem~\ref{1.smoothaff} in Chapter~\ref{affine}
(see \eqref{1.proddecomp} in \S 3 of that chapter).  
The general case follows by the gluing
construction.
\end{proof}

On the other hand, if $N \subseteq N'$ is a sublattice 
of finite index, then we have the following statement.

\begin{proposition}
\label{3.changelattice}
Let $N$ be a sublattice of finite index in $N'$, let
$\Sigma'$ be a fan in $N'_\R$ whose cones are generated
by elements of $N$.  Let $G = N'/N$.  Then 
\[
\varphi : X_{\Sigma',N} \to X_{\Sigma',N'}
\]
induced by the inclusion $\varphi : N \to N'$ presents
$X_{\Sigma',N'}$ as the quotient $X_{\Sigma',N}/G$.
\end{proposition}

\begin{proof}
By Proposition~\ref{1.quotient} of Chapter~\ref{affine}, 
since $N$ has finite index in $N'$, 
the finite group $G = N'/N$ acts on the affine open 
subset $U_{\sigma',N} \subseteq X_{\Sigma',N}$ for each 
cone in $\Sigma'$.  The coordinate ring of the affine
open subset $U_{\sigma',N'}$ is the subring of $G$-invariants
in the coordinate ring of $U_{\sigma',N}$, and hence $U_{\sigma',N'}$ 
is isomorphic to the quotient variety $U_{\sigma',N}/G$.  
The actions by $G$ on the affine opens are all compatible and glue 
together to give an action of $G$ on $X_{\Sigma',N}$.  It follows that
$X_{\Sigma',N'}$ is the quotient variety.
\end{proof}

\begin{example}
\label{3.P112quot}

Let $N' \simeq \Z^2$, and $\Sigma'$ be the fan shown in Figure~5,
so $X_{\Sigma',N'}$ is isomorphic to the weighted projective
space  $\P(1,1,2)$.  Let $N$ be the sublattice of $N'$ given by 
$N = \{(a,b) \in N' \mid b \equiv 0 \bmod 2\}$, so $N'$
has index 2 in $N$.
This $N'$ is generated by $v_1 = e_1$, $v_2 = 2e_2$.  Note that
\[
v_0 = -e_1 - 2e_2 = -v_1 - v_2 \in N.
\]  
Let $\overline{\varphi} : N \to N'$
be the inclusion map.    It is not difficult to see that with respect 
to the lattice $N$, $X_{\Sigma',N} \simeq \P^2$ 
(see Exercise~\ref{3.P112quotcheck}).  By 
Theorem~\ref{3.mapcorresp}, the $\Z$-linear map $\overline{\varphi}$
induces a toric morphism $\varphi : \P^2 \to \P(1,1,2)$, and
from Proposition~\ref{3.changelattice}, it follows that $\P(1,1,2)\simeq\P^2/G$.  

\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.7in]{ch3fig6.ps}
\end{center}
\caption{The semigroups $\sigma_2^\vee \cap M$ and $\sigma_2^\vee \cap M'$} 
\end{figure}
It is instructive to consider how the dual cone $\sigma_2^\vee$ 
for $\sigma_2$ in Figure~5 interacts with the lattice $M$ dual to $N$.
It is easy to check that $M \simeq \{(a,b/2) : a,b \in \Z\}$,
and $\sigma_2^\vee = \Cone(2e_1 - e_2, -e_2)$.  In Figure~6, the 
points in $\sigma_2^\vee\cap M'$ are shown in white, and 
the points in $\sigma_2^\vee\cap M$
not in $\sigma_2^\vee\cap M'$ are shown
in black.   Note that the picture in $\sigma_2^\vee\cap M'$
is the same (up to a change of coordinates in $\mathrm{GL}(2,\Z)$)
as that shown in Figure~11 from Chapter~\ref{affine}. This shows
again that $\P(1,1,2)$ contains the affine open subset $U_{\sigma_2,N}$
isomorphic to the rational normal cone $\widehat{C}_2$.  On the other
hand $U_{\sigma_2,N'} \simeq \C^2$ is smooth.  The other
affine open subsets corresponding to $\sigma_1$ and $\sigma_0$ are
isomorphic to $\C^2$ in both $\P^2$ and in $\P(1,1,2)$.
\eoe
\end{example}

In general, given any sublattice $N \subseteq N'$, a basis for $N$
can be completed to a basis for a sublattice $N''$ of finite index in $N'$.
Given a fan $\Sigma$ in $N_\R$, Propositions \ref{3.splitsublattice}
and \ref{3.changelattice} show that the toric variety $X_{\Sigma,N'}$
will be a quotient of the toric variety 
\[
X_{\Sigma, N''} \simeq X_{\Sigma,N} \times (\Cstar)^{n-k}
\]
by the finite abelian group $N'/N''$.
We leave it to the reader to construct examples of this general 
situation.  

\subsection*{\emph{Refinements of fans and blow-ups}}
Next, let $\Sigma, \Sigma'$
be two fans in the same lattice $N_\R$, but 
assume $\Sigma'$ is a \emph{refinement}
of $\Sigma$ (that is, for every cone $\sigma'$ in $\Sigma'$,
there exists $\sigma$ in $\Sigma$ such that $\sigma' \subseteq \sigma$.
In this case, the identity mapping on $N$ is automatically compatible
with $\Sigma$ and $\Sigma'$ and we obtain a toric morphism
$\varphi: X_{\Sigma'} \to X_\Sigma$.

\begin{example}
\label{3.refinefan}
Consider the fan $\Sigma'$ in $N\simeq \Z^2$ pictured in 
Figure~1 fom \S 1 of this chapter.  This is a refinement of the fan $\Sigma$ 
consisting of $\Cone(e_1,e_2)$ and its faces.  The 
corresponding toric varieties are $X_\Sigma \simeq \C^2$ and 
$X_{\Sigma'} \simeq W$, the blow-up of $\C^2$ at the origin
(see Example~\ref{3.blowupptfan}).  The toric
morphism induced by the identity on $N$ is 
$\varphi : W \to \C^2$, the ``blow-down'' morphism
that maps $\P^1 \times \{(0,0)\} \subseteq W$
to $(0,0) \in \C^2$ and is injective on the complement
of $\P^1 \times \{(0,0)\}$ in $W$.
\eoe
\end{example}

\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.8in]{ch3fig7.ps}
\end{center}
\caption{The star subdivision $\Sigma^*(\sigma)$} 
\end{figure}
We can generalize this example and Example~\ref{3.blowuporigin} from
\S 1 as follows.

\begin{definition}
\label{3.starsubdivision}
Let $\Sigma$ be a fan in $N_\R$ where $N\simeq \Z^n$.
Let $\sigma = \Cone(v_1,\ldots,v_n)$ be a smooth cone in $\Sigma$ 
(that is, we require that  $\{v_1,\ldots,v_n\}$ is a basis for $\Z^n$).  
Let $v_0 = v_1 + \cdots + v_n$ and let $\Sigma'(\sigma)$
be the fan consisting of all cones generated by subsets of
$\{v_0,\ldots,v_n\}$ not containing $\{v_1,\ldots,v_n\}$.
Let 
\[
\Sigma^*(\sigma) = \left(\Sigma \setminus \{\sigma\}\right) \cup \Sigma'(\sigma).
\]
Then $\Sigma^*(\sigma)$ is a fan in $N_\R$, called the \emph{\bfseries star subdivision}
of $\Sigma$ along $\sigma$.
\end{definition}
  
Figure~7 shows the star subdivision of a three-dimensional cone 
$\sigma = \Cone(v_1,v_2,v_3)$ into
three cones $\Cone(v_0,v_1,v_2)$, $\Cone(v_0,v_1,v_3)$, and 
$\Cone(v_0,v_2,v_3)$.  The fan $\Sigma^*(\sigma)$ consists of 
these, together with their faces.

\begin{proposition}
\label{3.starblowup}
With all notation as above, the 
toric variety $X_{\Sigma^*(\sigma)}$ is the blow-up of $X_\Sigma$
at the distinguished point $\gamma_\sigma$ corresponding to the 
cone $\sigma$, and we have a toric morphism 
\[
\varphi : X_{\Sigma^*(\sigma)} \to X_\Sigma.
\]  
\end{proposition}

\begin{proof}
Since $\Sigma$ and $\Sigma^*(\sigma)$ are the same outside the 
cone $\sigma$, without loss of generality, we may reduce to the 
case that $\Sigma$ is the fan consisting of $\sigma$ and all of
its faces, and $X_\Sigma$ is the affine toric variety 
$U_\sigma \simeq \C^n$.

Under the Orbit-Cone Correspondence (Theorem~\ref{3.occorresp}), 
$\sigma$ corresponds to the distinguished point $\gamma_\sigma$, 
the origin (a fixed point for the torus action).  By 
Theorem~\ref{3.mapcorresp}, the identity map on $N$ induces 
a toric morphism 
\[
\varphi : X_{\Sigma^*(\sigma)} \to U_\sigma \simeq \C^n.
\]  
It is easy to check that the affine open sets covering
$X_{\Sigma^*(\sigma)}$ are the same, and are glued together in
the same way, as those for the blow-up of $\C^n$ at the origin
from Exercise~\ref{3.blowups}.
In Exercise~\ref{3.blowuporigin} you will verify these claims.
\end{proof}

Even more generally, for any cone $\tau$ in $\Sigma$ generated by 
a subset of a basis for the lattice $N$ there exists a toric
variety $X_{\Sigma^*(\tau)}$ isomorphic to the blow-up of $X_\Sigma$ along the orbit
closure $\overline{O(\tau)}$ and a toric morphism 
\[
\varphi : X_{\Sigma^*(\tau)} \to X_\Sigma.
\]
The construction is a
more or less direct generalization of the construction of
$\Sigma^*(\sigma)$ as above; see \cite[Proposition 1.26]{oda}.

In Chapters~\ref{surface} and \ref{resolution}, we will also
see that toric blow-ups can be used to resolve singularities
of toric varieties.

\subsection*{\emph{Exact Sequences and Fibrations}}
Next, we will consider some of the local structure of toric
morphisms.  To begin, consider a surjective $\Z$-linear mapping
\[
\overline{\varphi} : N \twoheadrightarrow N'.
\]
If $\Sigma$ in $N_\R$ and $\Sigma'$ in $(N')_\R$ are compatible
with $\varphi$, then we have a corresponding toric morphism 
\[
\varphi : X_\Sigma \to X_{\Sigma'}.
\]
It is easy to check that 
\[
\Sigma_0 = \{\sigma \in \Sigma \mid \sigma \subseteq \ker(\overline{\varphi})\}
\]
is a fan in $N_\R$ (a subfan of $\Sigma$), and that $\overline{\varphi}$
is compatible with $\Sigma_0$ and the trivial fan $\{0\}$ in $N'$.  
This gives the toric morphism 
\[
\varphi\res{X_{\Sigma_0}} : X_{\Sigma_0} \to T_{N'}.
\]
In fact, by the reasoning used in the proof of
Proposition~\ref{3.mapcorresp}, under $\varphi : X_\Sigma \to X_{\Sigma'}$,
\begin{equation}
\label{3.invimtor}
\varphi^{-1}(T_{N'}) = X_{\Sigma_0}.
\end{equation}

The cones from $\Sigma_0$ also define a fan in the subspace
$(\ker(\varphi))_\R \subseteq N_\R$.  Call this fan $\Sigma''$.
Moreover, it is always true that a basis for $\ker(\varphi)$ 
can be extended to a basis of $N$ in this situation.  
By Proposition~\ref{3.splitsublattice}, 
\begin{equation}
\label{3.prodtor}
X_{\Sigma_0} \simeq X_{\Sigma''} \times (\Cstar)^{n-k} \simeq X_{\Sigma''} \times T_{N'}.
\end{equation}
Combining \eqref{3.invimtor} and \eqref{3.prodtor}, 
\[
\varphi^{-1}(T_{N'}) \simeq X_{\Sigma''} \times T_{N'}.
\]
In other words, over the affine open subset $T_{N'}$ in 
$X_{\Sigma'}$, via the mapping $\varphi$,
$X_\Sigma$ is identified with \emph{the product} of $T_{N'}$
and the toric variety $X_{\Sigma''}$.  We say this subset of $X_\Sigma$ is a 
\emph{fiber bundle} over $T_{N'}$ with fiber $X_{\Sigma''}$.
When the fan $\Sigma$ has a suitable structure, we can make a 
similar statement \emph{globally}.

Let $N'',N,N'$ be lattices and let
\begin{equation}
\label{3.exseq}
0 \longrightarrow N'' \stackrel{\overline{\psi}}{\longrightarrow} 
N \stackrel{\overline{\varphi}}{\longrightarrow} N' \longrightarrow 0
\end{equation}
be an exact sequence of $\Z$-linear maps. Let $\Sigma''$, $\Sigma$, and
$\Sigma'$ be fans in $(N'')_\R$, $N_\R$, and $(N')_\R$ respectively.  Assume
the mappings $\overline{\varphi}$ and $\overline{\psi}$ are compatible
with the corresponding fans.  We can view the cones in $\Sigma''$ as cones in 
$\overline{\psi}_\R((N'')_\R) \subseteq N_\R$.  

\begin{definition}
\label{3.splitfan}
In the situation of \eqref{3.exseq}, 
we say $\Sigma$ is \emph{\bfseries split by} $\Sigma'$ and $\Sigma''$
if there exists a fan $\widehat{\Sigma'}$ in $N_\R$ such that 
\begin{thmenum}
\item $\overline{\varphi}$
maps each cone in $\widehat{\Sigma'}$ bijectively to a cone in $\Sigma'$, and
\item each cone $\sigma$ in $\Sigma$ has the form $\sigma = \sigma' + \sigma''$
for some cone $\sigma'$ in $\widehat{\Sigma'}$ and $\sigma''$ in $\Sigma''$.
\end{thmenum}
\end{definition}

\begin{theorem}
\label{3.fibration}
Given an exact sequence \eqref{3.exseq}, let $\Sigma$ be a fan in $N_\R$
split by fans $\Sigma'$ in $(N')_\R$ and $\Sigma''$ in $(N'')_\R$.  
Then $X_\Sigma$ is a locally trival fiber bundle over $X_{\Sigma'}$ with
fiber $X_{\Sigma''}$.  That is,
$X_{\Sigma'}$ has a cover by open affine subsets $U$ with the
property that
\[
\varphi^{-1}(U) \simeq U \times X_{\Sigma''}.
\]
\end{theorem}

\begin{proof}
By condition a in Definition~\ref{3.splitfan}, we can identify
$\Sigma''$ with the set of cones in $\Sigma$ that map to the
trivial cone $\{0\}$ in $N'$.  As above, we have
\[
\varphi^{-1}(T_{N'}) \simeq T_{N'} \times X_{\Sigma''}.
\]
Then for all $\sigma'$ in $\Sigma'$,
by condition b in Definition~\ref{3.splitfan}, we identify
$\{\sigma \mid \overline{\varphi}(\sigma) \subseteq \sigma'\}$ with
\begin{equation}
\label{3.invimcones}
\Sigma(\sigma') = \{\sigma' + \sigma'' \mid \sigma'' \text{ in } \Sigma''\}.
\end{equation}
This collection of cones in $N$ also forms a fan, and the corresponding
toric variety maps to $U_{\sigma'} \subseteq X_{\Sigma'}$ under
the restriction of the toric morphism $\varphi$.  Hence because of the 
form of the cones in \eqref{3.invimcones}, it follows that
\[
\varphi^{-1}(U_{\sigma'}) = X_{\Sigma(\sigma')} \simeq U_\sigma' \times X_{\Sigma''},
\]
and the theorem is proved.
\end{proof}

\begin{example}
\label{3.HP1Ex3}
To complete the discussion from Examples \ref{3.HP1Ex1} and 
\ref{3.HP1Ex2}, consider the toric morphism $\varphi : \cH_r \to \P^1$
induced by the mapping 
\[
\overline{\varphi} : N \to N', \qquad ae_1 + be_2 \mapsto af.
\]
The fan $\Sigma$ in $N$ defining $\cH_r$ is split by $\Sigma'$
and $\Sigma''$ in this case because we have a fan
$\widehat{\Sigma'}$ in $N_\R$ consisting of 
\[
\Cone(-e_1 + re_2), \{0\}, \Cone(e_1),
\]
whose cones are mapped bijectively to the cones in $\Sigma'$
under $\varphi_\R$,  and $\Sigma''$ consisting of 
\[
\Cone(e_2), \{0\}, \Cone(-e_2).
\]
Each cone in $\Sigma$ has the form $\sigma' + \sigma''$ for 
$\sigma' \in \widehat{\Sigma'}$ and $\sigma'' \in \Sigma''$.
Hence Theorem~\ref{3.fibration} shows that $\cH_r$ is a 
locally trivial fibration over $\P^1$, with fibers 
isomorphic to 
\[
X_{\Sigma''} \simeq \P^1.
\]
If $r > 0$, then this fibration is not globally trivial.
That is, it is not true that $\cH_r \simeq \P^1 \times \P^1$.
There is some ``twisting'' on the fibers involved in the gluing 
of the $U_i \times \P^1$ in the construction of $\cH_r$.
\eoe
\end{example}

We will give another, more precise, description of
these fiber bundles and the ``twisting'' mentioned above
using the language of sheaves in 
Chapter~\ref{invertible}.

\exercises

\small

\begin{exercise}
\label{3.gluemaps}
Let $X$ be a variety with an affine open cover $\{U_i\}$, and
let $Y$ be a second variety.  Let $\varphi_i : U_i \to Y$ be
a collection of morphisms.  We say that a morphism
$\varphi : X \to Y$ is obtained by gluing the $\varphi_i$ 
if $\varphi\res{U_i} = \varphi_i$ for all $i$.
Show that there exists such a $\varphi : X \to Y$ if 
and only if for every pair $i,j$, 
\[
\varphi_i\res{U_i\cap U_j} = \varphi_j\res{U_i\cap U_j}.
\]
\end{exercise}

\begin{exercise}
\label{3.toricgluemaps}
Let $N, N'$ be lattices, and let $\Sigma$ in $N_\R$ and $\Sigma'$
in $(N')_\R$ be fans.  Let $\overline{\varphi} : N \to N'$ be
a $\Z$-linear mapping that is compatible with the corresponding fans.
Using Exercise~\ref{3.gluemaps} above, show that the induced toric 
morphisms $\varphi_\sigma : U_\sigma \to U_\sigma'$ glue together
to form a morphism $\varphi : X_\Sigma \to X_{\Sigma'}$.
\end{exercise}


\begin{exercise}
\label{3.P112quotcheck}
This exercise asks you to verify some of the claims made in 
Example~\ref{3.P112quot}.
\begin{thmenum}
\item  Verify that with respect to the lattice $N'$, $X_{\Sigma,N'} \simeq \P^2$.
\item  Verify carefully that the affine open subset
$U_{\sigma_2,N} \simeq \widehat{C}_2$, the rational normal cone of degree 2.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{3.blowupP2}
Let $\Sigma'$ be the fan obtained from the fan $\Sigma$ for $\P^2$ 
in Example~\ref{3.P2} by the following process.  Subdivide 
the cone $\sigma_2$ into two new cones $\sigma_{21}$ and $\sigma_{22}$
by inserting an edge $\Cone(-e_2)$.
\begin{thmenum}
\item Construct an affine open cover for $X_{\Sigma'}$ explicitly
and give the gluing homomorphisms.
\item Show that the resulting toric variety $X_{\Sigma'}$ is smooth.
\item Show that $X_{\Sigma'}$ is isomorphic to the Hirzebruch surface
$\cH_1$.  
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{3.blowupwproj}
Let $\Sigma'$ be the fan obtained from the fan $\Sigma$ for $\P(1,1,2)$ 
in Example~\ref{3.weightedproj} by the following process.  Subdivide 
the cone $\sigma_2$ into two new cones $\sigma_{21}$ and $\sigma_{22}$
by inserting an edge $\Cone(-v_2)$.
\begin{thmenum}
\item Construct an affine open cover for $X_{\Sigma'}$ explicitly
and give the gluing homomorphisms.
\item Show that the resulting toric variety $X_{\Sigma'}$ is smooth.
\item Construct a morphism $\phi : X_{\Sigma'} \to X_\Sigma$ and 
identify the fiber over the singular point of $X_\Sigma$.  
\item One of our smooth examples is isomorphic to $X_{\Sigma'}$.
Which one is it?
\end{thmenum}
\end{exercise}

%\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}

\normalsize

\section{Complete and Proper}

\subsection*{\emph{\bfseries The Compactness Criterion}}
In this section we will begin by proving part c of 
Theorem~\ref{3.fantoricprops}:
\begin{coxitem}
\item \emph{The toric variety $X_\Sigma$ is compact in its 
classical topology if and only if $\Sigma$ is a complete fan 
(that is, $|\Sigma| = \bigcup_{\sigma \in \Sigma} \sigma = N_\R$). }
\end{coxitem}
We will give a relatively elementary direct 
topological argument first, then reinterpret the key ideas to introduce 
a more general property of toric morphisms.

\begin{proof}
One part of this is an immediate consequence of Proposition~\ref{3.1pslimitpts}
from \S 2, and we used this, for instance, in analyzing limit points of one-parameter
subgroups in $\P^2$ in Example~\ref{3.P2fanfrom1ps} in that section.  
Namely, if $X_\Sigma$ is compact, then $\lim_{s\to 0} \lambda^n(s)$
exists in $X_\Sigma$ for every one-parameter subgroup $\lambda^n$ of $T_N$, that
is, for every $n \in N$.  Because $X_\Sigma$ is the union of the affine open 
subsets $U_\sigma$ for
$\sigma$ in $\Sigma$, this implies
\[
\lim_{s\to 0} \lambda^n(s) \in U_\sigma \text{ for some $\sigma$.}
\]
But then Proposition~\ref{3.1pslimitpts} implies
that $n \in \sigma$.  Since every lattice point $n \in N$ is contained in one 
of the cones of the fan, $\Sigma$ is complete.  

For the other implication, since $X_\Sigma$ is separated 
(Theorem~\ref{3.toricfromfan}), $X_\Sigma$ is Hausdorff as a topological
space (Theorem~\ref{3.sepHaus}).  In fact more is true.  The classical
topology on each affine open set $U_\sigma$ is even a metric topology.  
So compactness of $X_\Sigma$ is equivalent to the Bolzano-Weierstrass property 
(the statement that every sequence of points in $X_\Sigma$ has a convergent 
subsequence).  

The proof will proceed by induction on $d = \dim_\R N_\R$.  In the 
case $d = 1$, the only complete fan $\Sigma$ is the fan in $\R$
pictured in Example~\ref{2.P1P1} from Chapter 2, and in 
Example~\ref{3.faninsubspace} of this chapter.
The corresponding toric variety is $X_\Sigma = \P^1$. This 
is homeomorphic to $S^2$, the two-dimensional sphere, hence is
compact.  

Now, assume the statement has been proved for all complete fans $\Sigma$
of dimension strictly less than $d$, and consider a complete fan
$\Sigma$ in $N_\R \simeq \R^d$.  Let $\gamma_k$ for $k \in \N$ be
a sequence of points in $X_\Sigma$.  We will show that there is
a convergent subsequence $\gamma_{k_\ell}$ in $X_\Sigma$.  

Since $\Sigma$ is consists of finitely many cones, without
loss of generality we may assume the sequence $\gamma_k$ 
lies entirely within one affine open subset $U_\sigma$.
Let $\tau$ be a face of $\sigma$ of dimension $\ge 1$.  
By Corollary~\ref{3.starfan}, the closure of the orbit $O(\tau)$ 
in $X_\Sigma$ is the toric variety $X_{\mathrm{Star}(\tau)}$
of dimension $\le d - 1$.  If $\Sigma$ is
complete, then it is easy to check that the fan $\mathrm{Star}(\tau)$
is also complete in $N(\tau)_\R$ (Exercise~\ref{3.starcomplete}).
If infinitely many terms in our sequence lie in $\overline{O(\tau)}$,
then the induction hypothesis implies that there is a convergent
subsequence in $\overline{O(\tau)}$.  Hence, without loss of generality again,
we may assume that our sequence lies entirely in the torus
$T_N \subseteq U_\sigma$, with $\sigma$ one of the maximal cones in 
$\Sigma$.  

The key ingredient of the proof will be the following 
sort of logarithm mapping
\begin{eqnarray}
\label{3.logmap}
L : T_N \simeq N \otimes_\Z \Cstar & \longrightarrow & N_\R\\
                n \otimes s &\longmapsto& n\otimes \log |s|\notag
\end{eqnarray}
from $T_N$ onto $N_\R$.  For more properties of this mapping,
see Exercise~\ref{3.polar} below.  

Since $\Sigma$ is complete, the same is true for the fan
consisting of the cones $-\sigma$, where $\sigma$ is in
$\Sigma$.  Hence, by passing to a subsequence,
we may assume that there is some $\sigma'$ in $\Sigma$
such that 
\[
L(\gamma_k) \in -\sigma'
\]
for all $k \in \N$.  However, since the copies of the torus 
$T_N$ in the $U_\sigma$ for all the maximal cones in $\Sigma$
are identified under the gluing in the construction of
$X_\Sigma$, and the gluing map restricts to the identity
on $T_N$, without loss of generality again, we can assume
that $\sigma = \sigma'$ in the above, 
and
\begin{equation}
\label{3.negcone}
\gamma_k \in T_N \subseteq U_\sigma \text{ and } L(\gamma_k) \in -\sigma
\end{equation}
for all $k\in \N$.
We must now interpret what this means in the intrinsic
characterization of points of toric varieties used in 
\S 2 and \S 3 of this chapter.  

Recall that points in $T_N \simeq N\otimes_Z \Cstar$ can be thought of as group
homomorphisms in $\mathrm{Hom}_\Z(M, \Cstar)$ via the
identification
\[
n \otimes s \longleftrightarrow (\gamma : m \mapsto s^{\langle m,n\rangle}).
\]
Moreover, any such group homomorphism restricts to a semigroup homomorphism
in $\mathrm{Hom}(M\cap \sigma^\vee,\C)$, hence yields a point $\gamma$ 
in $U_\sigma$.

Write $\sigma = \Cone(v_1,\ldots,v_p)$.  Then for any point 
$\gamma \leftrightarrow n \otimes s\in T_N \subseteq U_\sigma$, the condition
$L(\gamma) \in -\sigma$ as in \eqref{3.negcone} implies
\[
n \otimes \log |s| = -\sum_{i=1}^p a_i v_i \text{ for some } a_i \ge 0.
\]
Hence if $m \in M \cap \sigma^\vee$, 
\[
\log |\gamma(m)| = \langle m,n \rangle \log|s| = -\sum_{i=1}^p a_i \langle m, v_i\rangle.
\]
By definition of $\sigma^\vee$, $\langle m, v_i\rangle \ge 0$ for all $i$.  
Since the $a_i \ge 0$, this implies that 
\[
\log|\gamma(m)| \le 0 \Longleftrightarrow |\gamma(m)| \le 1  
\text{ for all } m \in M \cap \sigma^\vee.
\]

We apply this to our sequence of points $\gamma_k$.  Since the $\gamma_k$
are a sequence of mappings to the closed unit disk in 
$\C$, which is compact, there is a subsequence $\gamma_{k_\ell}$ 
which converges to some $\gamma \in \mathrm{Hom}(M\cap \sigma^\vee,\C) = U_\sigma$.
You will check the details of this final assertion in Exercise~\ref{3.finishpf}.
\end{proof}

\subsection*{\emph{\bfseries Proper Mappings}}
The property of compactness also has a 
relative version that is used most often in the 
theory of complex manifolds.  

\begin{definition}
\label{3.properdef1}
A continuous mapping $f : X \to Y$ is 
\emph{\bfseries proper} if the inverse image $f^{-1}(T)$
is compact in $X$ for every compact subset $T\subseteq Y$. 
\end{definition}

It is immediate that $X$ is compact if and only if the constant 
mapping from $X$ to the space $Y = \{pt\}$ consisting of a single point
is proper.  This relative version of compactness may also
be reformulated, for reasonably nice topological spaces,
in the following way (see \cite[Chapter 9, \S 4]{gr}).

\begin{proposition}
Let $f : X \to Y$ be a continuous mapping of locally compact
Hausdorff spaces.  Then $f$ is proper if and only if
$f$ is a closed mapping (that is, $f(U)$ is closed in $Y$
for all closed subsets $U \subseteq X$), and all its fibers
$f^{-1}(\{y\})$, $y \in Y$, are compact. \qed
\end{proposition}

Although this criterion for properness can be used to 
characterize the proper toric morphisms, it is somewhat
awkward because the fibers of general toric morphisms can 
be quite complicated.  
So we will look to another reformulation.  Recall from
\S 0 of this chapter that the fibers of a mapping can
be described using the fibered product:
if $f : X \to Y$ and $y \in Y$, then $f^{-1}(\{y\})$
is $X \times_Y \{y\}$ using the inclusion mapping
$\{y\} \hookrightarrow Y$.  In Exercise~\ref{3.ucequivprop}, you will show that 
for maps of locally compact Hausdorff spaces,
the following universal closure property is \emph{equivalent}
to properness as in Definition~\ref{3.properdef1}.

\begin{definition}
\label{3.uc}
A continuous mapping $f : X \to Y$ is said
to be \emph{\bfseries universally closed} if for all
spaces $Z$ and all continuous mappings $g : Z \to Y$, in the 
commutative diagram
\[
\SelectTips {cm}{}
\xymatrix{
X \times_Y Z \ar[r]^(.6){\pi_X} \ar[d]_{\pi_Z} & X \ar[d]^{f} \\
Z \ar[r]_{g} & Y
}
\]
the projection $\pi_Z$ is a closed mapping.
\end{definition}

Hence, in algebraic geometry, it is now customary to 
use the following definition of properness for morphisms
of algebraic varieties. 

\begin{definition}
\label{3.properdef2}
A morphism of varieties $\varphi : X \to Y$ is
\emph{\bfseries proper} if it is universally closed
in the sense that for all varieties $Z$ and
morphisms $\psi : Z \to Y$, in the commutative 
diagram 
\[
\SelectTips {cm}{}
\xymatrix{
X \times_Y Z \ar[r]^(.6){\pi_X} \ar[d]_{\pi_Z} & X \ar[d]^{\varphi} \\
Z \ar[r]_{\psi} & Y
}
\]
the projection $\pi_Z$ is a closed mapping in the Zariski topology.
A variety $X$ is said to be \emph{\bfseries complete}
if the constant morphism $\varphi : X \to \{pt\}$ is proper.
\end{definition}

\begin{example}
\label{3.completeandnot}
The Projective Extension Theorem \cite[Chapter 8, \S 5, Theorem 6]{iva}
shows that with $X = \P^n$, the mapping 
\[
\pi_{\C^m} : \P^n \times \C^m \to \C^m
\]
is closed in the Zariski topology for all $m$.  It follows that 
if $V \subseteq \C^m$ is any affine variety, the projection
\[
\pi_V : \P^n \times V \to V
\]
is a closed mapping in the Zariski topology.
By the gluing construction, 
the constant morphism $\P^n \to \{pt\}$
is proper, so $\P^n$ is a complete variety (the 
prototypical complete variety).  Moreover, any projective variety
is complete (Exercise~\ref{3.projcomp}).

On the other hand consider the morphism $\C \to \{pt\}$.
We claim that this is not proper, so $\C$ is not complete.  
To see this, consider
$\C \times_{\{pt\}} \C = \C^2$ and the diagram
\[
\SelectTips {cm}{}
\xymatrix{
\C^2 \ar[r]^{\pi_2} \ar[d]_{\pi_1} & \C \ar[d] \\
\C \ar[r] & \{pt\}
}
\]
The closed subset $\V(xy - 1) \subseteq \C^2$ does not
map to a Zariski-closed subset of $\C$ under $\pi_1$.
Hence $\pi_1$ is not a closed mapping, and $\C$ is not complete.
\eoe
\end{example}

Completeness is the algebraic version of compactness,
and it can be shown that complete varieties are compact
in the classical topology.  Similarly, proper morphisms
are proper maps as in Definition~\ref{3.properdef1}
when we consider the classical topology on the varieties
$X$ and $Y$.  

\subsection*{\emph{\bfseries The Properness Criterion}}
Part c of Theorem~\ref{3.fantoricprops} can be 
understood as a special case
of the following statement for toric morphisms.

\begin{theorem}
\label{3.propercriterion}
Let $\overline{\varphi}: N \to N'$ be a $\Z$-linear mapping
compatible with fans $\Sigma$ in $N_\R$ and $\Sigma' \in (N')_\R$.
The corresponding toric morphism
\[
\varphi : X_\Sigma \longrightarrow X_{\Sigma'}
\]
is proper if and only if 
\[
\overline{\varphi}_\R^{\,\,-1}(|\Sigma'|) = |\Sigma|.
\]
\end{theorem}

Before we can prove this, however, we need to introduce
yet another characterization of 
properness.  While Definition~\ref{3.properdef1} would
lead us too far into the specifics of toric morphisms,
Definition~\ref{3.properdef2} is too general
to be convenient as a way to prove a particular
morphism is proper.  As the second part of 
Example~\ref{3.completeandnot} indicates, in order for 
a morphism to be proper, we cannot have sequences of 
points on closed subsets of $X \times_Y Z$ 
``escaping to infinity.''  To test for that, it
suffices to consider what happens on curves.

\begin{theorem}[reference?] 
\label{3.ersatzvaluativecriterion}  A morphism of 
varieties $\varphi : X \to Y$ is proper if and only
if the following statement holds:  If 
$C$ is any smooth curve, $p \in C$
is a point, and $\lambda$ is any morphism 
\[
\lambda : C \setminus \{p\}  \longrightarrow X
\]
such that $\psi = \varphi \circ \lambda$ extends to a 
morphism $\overline{\psi} : C \to Y$, 
then $\lambda$ extends to a morphism $\overline{\lambda} : C \to X$.  
In other words, if $i$ is the inclusion 
$C \setminus \{p\} \hookrightarrow C$, then in any diagram
\[
\SelectTips {cm}{}
\xymatrix{
C \setminus \{p\} \ar[r]^(.6){\lambda} \ar[d]_{i} & X \ar[d]^{\varphi} \\
C \ar@{.>}[ur]_{\overline{\lambda}} \ar[r]_{\overline{\psi}} & Y
}
\]
where the outside square commutes, 
there exists an extension of $\lambda$ to 
$\overline{\lambda} : C \to X$ making the entire diagram
commute. \qed
\end{theorem}

Readers with some experience in the theory of schemes will recognize
this statement as a version of the {\it valuative criterion of properness}
(\cite[Theorem 4.7 in Chapter II]{hartshorne}).  Although we 
have provided the foregoing remarks as context and motivation, 
for our purposes, nothing vital would be lost by simply taking this
criterion as the \emph{definition} of properness.  

A smooth curve is a smooth variety of dimension 1
hence has the structure of a complex manifold
of dimension 1 (a Riemann surface).  By reducing to curves we can 
take advantage of the simple function theory
on these varieties.  In particular, at each point
of a smooth curve there is a local analytic coordinate
and we can represent functions by Laurent series. 
We are now ready to prove Theorem~\ref{3.propercriterion}.

\begin{proof}
First we will show that if $\varphi$ is proper, then 
$\overline{\varphi}_\R^{\,\,-1}(|\Sigma'|) = |\Sigma|$.  
By the definition of compatibility of $\overline{\varphi}$
with the fans, the inclusion 
\[
\overline{\varphi}_\R^{\,\,-1}(|\Sigma'|) \supseteq |\Sigma|
\]
is automatic.  Because all the cones are 
rational, it will suffice consider 
$n \in \overline{\varphi}_\R^{\,\,-1}(|\Sigma'|) \cap N$
to show the opposite inclusion.  Let $\overline{\varphi}(n) = n' \in \sigma'$
in $\Sigma'$, and consider the one-parameter subgroup
\[ 
\lambda^{n} : \Cstar \to T_{N} \subseteq X_{\Sigma}.
\]
This is a mapping from the curve $\Cstar = \C \setminus \{0\}$ into
$X_{\Sigma}$.  We have 
\[
\varphi(\lambda^n(s)) = \lambda^{\overline{\varphi}(n)}(s) = \lambda^{n'}(s) 
\]
for all $s \ne 0$.  

Since $n'\in \sigma'$, by Proposition~\ref{3.1pslimitpts},
$\lim_{s\to 0} \lambda^{n'}(s)$ exists in $U_{\sigma'} \subseteq X_{\Sigma'}$. 
This means that $\lambda^{n'}$ extends to a morphism 
\[
\overline{\lambda^{n'}} : \C \to X_{\Sigma'},
\]
and we have a commutative diagram as in the statement of 
Theorem~\ref{3.ersatzvaluativecriterion}:
\[
\SelectTips {cm}{}
\xymatrix{
\C \setminus \{0\} \ar[r]^(.6){\lambda^{n}} \ar[d]_{i} & X_\Sigma \ar[d]^{\varphi} \\
\C \ar@{.>}[ur]_{\overline{\lambda^n}} \ar[r]_{\overline{\lambda^{n'}}} & X_{\Sigma'}\ .
}
\]

Since we assume $\varphi$ is proper, there exists an extension
of $\lambda^n$, $\overline{\lambda^n} : \C \to X_\Sigma$ making the 
diagram commute.  It follows that $\lim_{s \to 0}\lambda^n(s)$ exists 
and equals a point in $X_\Sigma$.  
Hence $n \in \sigma$ for some $\sigma$ in $\Sigma$ by 
Proposition~\ref{3.1pslimitpts}.  

For the opposite implication, assume that 
$\overline{\varphi}_\R^{\,\,-1}(|\Sigma'|) = |\Sigma|$, and
assume we have a commutative 
diagram
\begin{equation}
\label{3.fillin}
\SelectTips {cm}{}
\xymatrix{
C \setminus \{p\} \ar[r]^(.6){\lambda} \ar[d]_{i} & X_\Sigma \ar[d]^{\varphi} \\
C \ar[r]_{\overline{\psi}} & X_{\Sigma'}
}
\end{equation}
where $C$ is any smooth curve and $p$ is a point of $C$.  
We must show that $\lambda$ extends to $\overline{\lambda} : C \to X_\Sigma$
making the diagram commute.  
Let $\sigma'$ be a cone in $\Sigma'$ such that 
$\overline{\psi}(p) \in U_{\sigma'}$.  Without loss of generality,
we can replace $X_{\Sigma'}$ by the affine toric variety $U_{\sigma'}$,
and we can also take $C$ affine 
such that $\lambda(C \setminus \{p\}) \subseteq T_N$ and replace
$X_\Sigma$ by $T_N$.  We want
to see that there exists some cone $\sigma$ in $\Sigma$ such that
we can define $\lambda(p) \in U_\sigma$ to extend $\lambda$ to $C$.

By our reductions, corresponding to the morphisms in \eqref{3.fillin},
we have a commutative diagram of $\C$-algebra homomorphisms
\begin{equation}
\label{3.fillinrev}
\SelectTips {cm}{}
\xymatrix{
\C[C \setminus \{p\}]  & \C[M] \ar[l]_(.4){\lambda^*} \\
\C[C]  \ar[u]_{i^*} & \C[(\sigma')^\vee\cap M'] \ar[l]^(.55){\overline{\psi}^{\,*}} 
\ar[u]_{\varphi^*}
}
\end{equation}
The fact that $\psi = \varphi \circ \lambda$ extends to $C$ means
that every element of the image of $\overline{\psi}^{\,\,*}$ is a function 
on $C$ with no pole at $p$.  We may use Laurent expansions in terms
of a local analytic coordinate at $p$ on $C$ and write $\mathrm{ord}_p$ 
for the order of pole or vanishing.  Then 
$\mathrm{ord}_p \circ \overline{\psi}^{\,\,*}$
gives a $\Z$-linear mapping from 
$\{\chi^{m'} : m' \in M'\} \simeq M'$ to $\Z$ that is nonnegative 
on $(\sigma')^\vee\cap M'$.  By duality, this is the same
as an element of $\mathrm{Hom}_\Z(M',\Z) = N'$ that lies
in $(\sigma')^{\vee\vee} = \sigma'$.  

Since $\overline{\varphi}_\R^{\,\,-1}(|\Sigma'|) = |\Sigma|$,
there exists $\sigma$ in $\Sigma$ such that 
$\overline{\varphi}_\R(\sigma) \subseteq \sigma'$.  Then 
the commutativity of \eqref{3.fillinrev} implies
that $\mathrm{ord}_p\circ\lambda^* \ge 0$ for all functions in
$\C[\sigma^\vee \cap M] \subseteq \C[M]$ also.  
This gives the existence of the extension of $\lambda$, 
and hence $\varphi$ is a proper morphism by Theorem~\ref{3.ersatzvaluativecriterion}.
\end{proof}

\begin{example}
\label{3.blowupproper}
An important class of examples of proper morphisms are the blowups
\[
\varphi : X_{\Sigma^*(\sigma)} \to X_\Sigma
\]
studied in Proposition~\ref{3.starblowup}.  In that case,
the hypotheses for Theorem~\ref{3.propercriterion} are satisfied
since $|\Sigma| = |\Sigma^*(\sigma)|$ (the fan $\Sigma^*(\sigma)$
is a refinement of $\Sigma$ with the same support), and $\overline{\varphi}$
is the identity on $N$.  We can also see that the resulting toric morphism
is proper.  For instance, in the blowup
\[
\varphi : \mathrm{Bl}_0(\C^2) \to \C^2,
\]
the inverse image of each point other than $(0,0) \in \C^2$ is a
single point, while the inverse image of $(0,0)$ is the compact
set $\P^1 \times {(0,0)} \subset \mathrm{Bl}_0(\C^2)$.  In this
case, it is not difficult
to check directly that 
$\varphi^{-1}(T)$ is compact for all compact $T$ in $\C^2$.  
\eoe
\end{example}
\exercises
\small

\begin{exercise}
Give a second argument for the implication
\[
X_\Sigma \text{ compact } \Rightarrow \Sigma \text{ complete }
\]
from part c of Theorem~\ref{3.fantoricprops}
using induction on the dimension $d$ of $N$.  Hint:  If $\Sigma$ is not 
complete and $d > 1$, then there is a one-dimensional cone $\tau$ 
in the boundary of the support of $\Sigma$.  Consider the fan
$\mathrm{Star}(\tau)$ and the corresponding toric subvariety
of $X_\Sigma$.  
\end{exercise}

\begin{exercise}
\label{3.starcomplete}
Let $\Sigma$ be a complete fan in $N_\R$.  Let $\tau$ be any 
cone in $\Sigma$ and let $\mathrm{Star}(\tau)$ be as in
\eqref{3.star}.  Show that $\mathrm{Star}(\tau)$ is a 
complete fan in $N(\tau)_\R$.
\end{exercise}

\begin{exercise}
\label{3.polar}
In this exercise, you will develop some additional properties
of the logarithm mapping from \eqref{3.logmap}.
\begin{thmenum}
\item Let $S^1$ be the unit circle in the complex plane, a 
subgroup of the multiplicative group $\Cstar$.  Show
that there is an isomorphism of groups 
\begin{eqnarray*}
\varphi : \Cstar & \longrightarrow & S^1 \times \R \\
            z & \longmapsto & (|z|,\log |z|),
\end{eqnarray*}
where the operation in the second factor on the right is addition.
\item Show that the compact real $n$-dimensional torus
$(S^1)^n$ can be viewed as a subgroup of $T_N$, and that
$T_N/(S^1)^n \simeq N_\R$.  Hint: Use $\varphi$ from part a.
\item Let $\Sigma$ be a fan in $N$.
Show that the action of the compact real torus
$(S^1)^n \subseteq T_N$ on $T_N$ extends to an action on 
the toric variety $X_\Sigma$, and that the quotient space
\[
(X_\Sigma)/(S^1)^n \cong
\bigcup_{\sigma} N(\sigma)_\R,
\]
where $\cong$ denotes homeomorphism of topological spaces, and 
the union is over all cones in the fan.  
Hint: Use the Orbit-Cone Correspondence (Theorem~\ref{3.occorresp}).  
\item Let $\Sigma$ in $\R^2$ be the fan from Example~\ref{3.P2},
so $X_\Sigma \simeq \P^2$.  Show that under the action of 
$(S^1)^2 \subseteq (\Cstar)^2$ as in part c,
$\P^2/(S^1)^2 \cong \Delta_2$, the 2-dimensional simplex.  
\end{thmenum}
We will say much more about the topology of toric varieties 
in Chapter~\ref{topology}.
\end{exercise}

\begin{exercise}
\label{3.finishpf}  
In this exercise, you will supply the details to complete the 
proof of part c of Theorem~\ref{3.fantoricprops} from the 
start of this section.  Assume that $\gamma_k \in \mathrm{Hom}(M\cap \sigma^\vee,\C)$
is a sequence of semigroup homomorphisms, and assume that 
$|\gamma_k(m)| \le 1$ for all $m \in M \cap \sigma^\vee$ and 
all $k \in \N$.  We want to show that there is a subsequence
$\gamma_{k_\ell}$ that converges to some $\gamma \in \mathrm{Hom}(M\cap\sigma^\vee,\C)$.  
\begin{thmenum}
\item  The semigroup $\sS_\sigma = M \cap \sigma^\vee$
is generated by a finite Hilbert basis $\{m_1,\ldots,m_s\}$.
Use this fact and the compactness of $\{z \in \C \mid |z| \le 1\}$
to show that there exists a subsequence $\gamma_{k_\ell}$ such 
that the sequences $\gamma_{k_\ell}(m_j)$ converge in $\C$ for all $j$.
\item Deduce that the subsequence $\gamma_{k_\ell}$ converges to 
a $\gamma \in \mathrm{Hom}(M\cap\sigma^\vee,\C)$.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{3.ucequivprop}
Show that a mapping of locally compact topological spaces
$f : X \to Y$ is proper if and only if it universally closed. 
Hint: Show first that compactness of $X$ is equivalent to the 
statement that the mapping $f : X \to \{pt\}$ is universally
closed.  Then use the easy fact that any composition of 
universally closed mappings is universally closed.
\end{exercise}

\begin{exercise}
\label{3.projcomp}
Show that any projective variety is complete according
to Definition~\ref{3.properdef2}.
\end{exercise}


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