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2000:\ Geometry of Toric Varieties} \strut}

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\medskip
\centerline{\bigbf Lecture 3}
\medskip
\centerline{\bigbf Projective Varieties and Abstract Varieties}
\medskip
\centerline{\bigrm David A. Cox}
\bigskip
\medskip

This lecture will review projective varieties in $\P^n$ and discuss
how abstract varieties are constructed by gluing affine varieties
together.  We conclude with some observations about gluing affine
toric varieties.
\bigskip
\medskip

\noindent {\bigbf \S1. Projective Varieties}
\bigskip
\smallskip

\noindent \ul{P\sc{rojective} S\sc{pace}}.  We define {\it $n$-dimensional
projective space\/} to be the set
$$
\P^n = (\C^{n+1}-\{0\})/\!\sim,
$$
where $\sim$ is the equivalence relation on $\C^{n+1}-\{0\}$ given by
$$
(a_0,\ldots,a_n) \sim (b_0,\ldots,b_n) \iff \hbox{there is } \lambda
\in \C^*\ \hbox{with}\ (a_0,\ldots,a_n) =
\lambda(b_0,\ldots,b_n).\leqno(1.1)  
$$
Here, we use $\C^*$ to denote $\C-\{0\}$, which is a group under
multiplication.  As we vary $\lambda \in \C^*$, the points
$\lambda(b_0,\ldots,b_n)$ lie on a line through the origin.  Thus we
get a bijection
$$
\P^n \simeq \{ \hbox{lines through the origin in $\C^{n+1}$}\}.
$$
\smallskip

\ex{1.1} $\P^n$ contains the subset $(\C^*)^{n+1}/\!\sim$.  Note also
that $(\C^*)^{n+1}$ is a group under component-wise multiplication.
\item{a.} Show that on $(\C^*)^{n+1}$, the equivalence classes of
$\sim$ are the cosets of the subgroup $H = \{(\lambda,\ldots,\lambda)
\mid \lambda \in \C^*\} \subset (\C^*)^{n+1}$.  Conclude that
$(\C^*)^{n+1}/H \subset \P^n$.
\item{b.} Construct a group isomorphism $(\C^*)^{n+1}/H \simeq
(\C^*)^n$.
\bigskip

Exercise 1.1 shows that $\P^n$ contains an isomorphic copy of
$(\C^*)^n$.  In Lecture 2, we saw that every $n$-dimensional affine
toric variety contains the torus $(\C^*)^n$.  We will see in Lecture 4
that $\P^n$ is a classic example of a non-affine toric variety.

We note that $\P^n$ has a {\it classical topology\/} inherited from
the usual topology on $\C^{n+1}-\{0\}$.
\bigskip

\ex{1.2} Let $S^{2n+1}$ be the unit $(2n+1)$-sphere centered at the
origin in $\C^{n+1}$.
\item{a.} Show that the natural map $S^{2n+1} \to \P^n$ is onto and
conclude that $\P^n$ is compact.
\item{b.} Show that the fibers of $S^{2n+1} \to \P^n$ are isomorphic
to $S^1$.  This is the {\it Hopf fibration\/}.
\bigskip
\smallskip

\noindent \ul{H\sc{omogeneous} C\sc{oordinates}}. A point $p$ of
$\P^n$ will be written $(a_0,\ldots,a_n)$.  This is only unique up to
the equivalence relation (1.1).  We call $(a_0,\ldots,a_n)$ {\it
homogeneous coordinates\/} of $p$.  In some books, this is written $p
= [a_0,\ldots,a_n]$ or $p = (a_0\!:\!\ldots\!:\!a_n)$ to emphasize the
non-unique nature of these coordinates.  We prefer to write $p =
(a_0,\ldots,a_n)$, where it will be clear from the context that we are
using homogeneous coordinates.

\smallskip
\bigskip
\noindent \ul{P\sc{rojective} V\sc{arieties and} H\sc{omogeneous}
I\sc{deals}}.  A polynomial $f \in \C[x_0,\ldots,x_n]$ is {\it
homogeneous of degree\/} $d$ if every term of $f$ has total degree
$d$.  This is equivalent to the identity 
$$
f(\lambda x_0,\ldots, \lambda x_n) = \lambda^d
f(x_0,\ldots,x_n). \leqno(1.2) 
$$
\smallskip

\ex{1.3} Show that any $f \in \C[x_0,\ldots,x_n]$ can be written
uniquely in the form $f = \sum_{d\ge 0} f_d$ where $f_d$ is
homogeneous of degree $d$.  We call $f_d$ the {\it homogeneous
components\/} of $f$.
\bigskip

Now suppose that $f\in \C[x_0,\ldots,x_n]$ is homogeneous of degree
$d$.  Given $p \in \P^n$, we can't define ``$f(p)$'' since using $p =
(a_0,\ldots,a_n)$ would give
$$
f(p) = f(a_0,\ldots,a_n),
$$
while using $p = \lambda (a_0,\ldots,a_n)$ would give 
$$
f(p) = f(\lambda a_0,\ldots, \lambda a_n) = \lambda^d
f(a_0,\ldots,a_n).
$$
However, the equation $f(p) = 0$ is well-defined since $\lambda \in
\C^*$.  Thus, homogeneous polynomials $f_1,\ldots,f_s \in
\C[x_0,\ldots,x_n]$ define the {\it projective variety\/}
$$
{\bf V}(f_1,\ldots,f_s) = \{a \in \P^n \mid f_1(a) = \cdots = f_s(a) =
0\} \subset \P^n.
$$

To formulate this in terms of ideals, we say that an ideal $I \subset
\C[x_0,\ldots,x_n]$ is {\it homogeneous\/} if it is generated by
homogeneous polynomials.
\bigskip

\ex{1.4} Show that an ideal $I \subset \C[x_0,\ldots,x_n]$ is
homogeneous if and only if for all $f\in\C[x_0,\ldots,x_n]$, we have
$f \in I \Leftrightarrow I$ contains the homogeneous components 
of $f$.
\bigskip

If $I \subset \C[x_0,\ldots,x_n]$ is a homogeneous ideal, then we
have the projective variety
$$
{\bf V}(I) = \{a \in \C^n \mid f(a) = 0\ \hbox{for all}\ f \in I\}.
$$
Conversely, given an projective variety $V \subset \C^n$, we get the
homogeneous ideal 
$$
{\bf I}(V) = \{f \in \C[x_0,\ldots,x_n] \mid f(a) = 0\ \hbox{for all}\
a \in V\}.
$$
\smallskip

\ex{1.5} We call $\langle x_0,\ldots,x_n\rangle \subset
\C[x_0,\ldots,x_n]$ the {\it irrelevant ideal\/}.  Show that ${\bf
V}(I) = \emptyset$ whenever $I$ contains a power of the irrelevant
ideal. 
\bigskip

Exercise 1.5 is actually part of the projective version of the
Nullstellensatz, which goes as follows.  We refer the reader to [1,
Chapter 8] for a proof.

\proclaim Projective Nullstellensatz.  Let $I \subset
\C[x_0,\ldots,x_n]$ be a homogeneous ideal.
\item{a.} ${\bf V}(I) = \emptyset$ if and only if $\langle
x_0,\ldots,x_n\rangle^m \subset I$ for some $m \ge 0$.
\item{b.} ${\bf V}(I) \ne \emptyset$ implies ${\bf I}({\bf V}(I)) = 
\sqrt{I}$. 

Most of the concepts defined for affine varieties in $\C^n$ can be
extended to projective varieties in $\P^n$ in the obvious way:
\item{$\bullet$} $W \subset V$ is a {\it subvariety\/} of a projective
variety $V \subset \P^n$ if $W$ is a projective variety in $\P^n$.  
\item{$\bullet$} If $V \subset \P^n$ is a projective variety, then we
call $\P^n - V$ a {\it Zariski open\/} subset of $\P^n$.
\item{$\bullet$} The {\it Zariski topology\/} is the topology on
$\P^n$ whose open sets are the Zariski open sets.
\item{$\bullet$} The {\it Zariski closure\/} $\overline{S}$ of a
subset $S\subset\P^n$ is the smallest projective variety containing
$S$. 

\smallskip
\bigskip
\noindent \ul{R\sc{ational} F\sc{unctions on} P\sc{rojective}
S\sc{pace}}.  We've already seen that a homogeneous polynomial in
$\C[x_1,\ldots,x_n]$ does not give a function on $\P^n$.  However, the
quotient of two such polynomials works, provided they have the same
degree.  More precisely, suppose that $f,g \in \C[x_1,\ldots,x_n]$
have degree $d$ and that $g \ne 0$.  Then (1.2) shows that we get a
well-defined function
$$
{f\over g} : \P^n -{\bf V}(g) \longrightarrow \C
$$
As in Lecture 1, we write this as $f/g : \P^n -\!\to \C$ and 
say that $f/g$ is a {\it rational function\/} on $\P^n$.
\bigskip

\ex{1.6} The set of all rational functions on $\P^n$ is
$$
\C(\P^n) = \Big\{{f\over g} \in \C(x_0,\ldots,x_n) \mid f,g\ \hbox{are
homogeneous of degree}\ d\ \hbox{and}\ g \ne 0\Big\}.
$$
Prove that $\C(\P^n)$ is subfield of $\C(x_0,\ldots,x_n)$.
\bigskip
\smallskip

\noindent \ul{M\sc{appings} B\sc{etween} P\sc{rojective}
V\sc{arieties}}. Suppose that $V \subset \P^n$ is a projective variety
and $f_0,\ldots,f_m \in \C[x_0,\ldots,x_n]$ are homogeneous
polynomials all of the same degree.  Then we say that $f_0,\ldots,f_m$
have no {\it base points\/} on $V$ if $V \cap {\bf V}(f_0,\ldots,f_m)
= \emptyset$.  
\bigskip

\ex{1.7} Suppose that $f_0,\ldots,f_m \in \C[x_0,\ldots,x_n]$ are
homogeneous of degree $d$ and have no base points on $V$.  Prove that
the $(a_0,\ldots,a_n) \mapsto
(f_0(a_0,\ldots,a_n),\ldots,f_m(a_0,\ldots,a_n))$ induces a
well-defined function $F : V \longrightarrow \P^m$.
\bigskip

An important fact is that in the situation of Exercise 1.6, the image
$F(V) \subset \P^m$ is a projective subvariety.  When $V = \P^n$, this
is proved in [1, Chapter 8], and the proof extends easily to cover the
general case.
\bigskip

\ex{1.8} When $V \subset \C^n$ and $F : V \to \C^m$ is a polynomial
map, the image $F(V) \subset \C^m$ need not be a subvariety.  For
example, suppose that $V = {\bf V}(xy-1) \subset \C^2$ and $F: V \to
\C$ is $F(x,y) = x$.  Prove that $F(V)$ is not a subvariety of $\C$.
The fact that $F(V)$ is a subvariety in the projective case is one
reason why projective varieties are so useful in algebraic geometry.
\bigskip
\smallskip

\noindent \ul{A\sc{ffine} O\sc{pen} S\sc{ubsets}}.  We can regard 
$\P^n$ as a union of affine spaces as follows.  For $0 \le i \le n$,
consider the Zariski open set $U_i = \P^n - {\bf V}(x_i)$.
\bigskip

\ex{1.9} As above, $U_i = \P^n - {\bf V}(x_i) = \{(a_0,\ldots,a_n) \in
\P^n \mid a_i \ne 0\}$.
\item{a.} Show that $U_i \simeq \C^n$ via $(a_0,\ldots,a_n) \mapsto
(a_0/a_i,\ldots,a_{i-1}/a_i,a_{i-1}/a_i,\ldots,a_n/a_i)$.
\item{b.} Show that ${\bf V}(x_i) \simeq \P^{n-1}$ via
$(a_0,\ldots,a_n) \mapsto (a_0,\ldots,a_{i-1},a_{i+1},\ldots,a_n)$.
\item{c.} Show that $\P^n = U_0 \cup\cdots\cup U_n$.
\bigskip

This exercise shows that we can regard $\P^n$ as $\C^n$ together with
a copy of $\P^{n-1}$ ``at infinity''.  Also, the open cover of
Exercise~1.9 shows that projective varieties are unions of affine
varieties.
\bigskip

\ex{1.10} Let $V = {\bf V}(f_1,\ldots,f_s) \subset \P^n$ be a
projective variety.  Prove that under the map $U_i \simeq \C^n$ from
Exercise~1.9, $V\cap U_i$ corresponds to an affine variety defined by
the vanishing of $\tildef_j$, where
$\tildef_{\!j}(x_0,\ldots,x_{i-1},x_{i+1},\ldots,x_n) =
f_j(x_0,\ldots,x_{i-1},1,x_{i+1},\ldots,x_n)$.  We call
$\tildef_{\!j}$ the {\it dehomogenization\/} of $f_j$ with respect to
$x_i$.
\bigskip

Another way to think about $U_i \simeq \C^n$ is to use
$x_0/x_i,\ldots,x_{i-1}/x_i,x_{i-1}/x_i,\ldots,x_n/x_i$ as variables
on $\C^n$.  Then the dehomogenization map of Exercise 1.10 is just $f
\mapsto f/x_i^d$, where $f \in \C[x_0,\ldots,x_n]$ is homogeneous of
degree $d$.  This approach preserves rational functions.
\bigskip

\ex{1.11} Show that the map $f/g \mapsto (f/x_i^d)/(g/x_i^d)$ 
induces an isomorphism of fields
$$
\C(\P^n) \simeq
\C(x_0/x_i,\ldots,x_{i-1}/x_i,x_{i-1}/x_i,\ldots,x_n/x_i),
$$
where $\C(\P^n)$ is the field of rational functions defined in
Exercise~1.6.
\bigskip
\smallskip

\noindent \ul{W\sc{eighted} P\sc{rojective} S\sc{pace}}.  We next
discuss a generalization of $\P^n$.  Given positive integers
$q_0,\ldots,q_n$ satisfying ${\rm gcd}(q_0,\ldots,q_n) = 1$, we get
the {\it weighted projective space\/}
$$
\P(q_0,\ldots,q_n) = (\C^{n+1}-\{0\})/\!\sim,
$$
where $\sim$ is the equivalence relation on $\C^{n+1}-\{0\}$ given by
$$
(a_0,\ldots,a_n) \sim (b_0,\ldots,b_n) \iff \hbox{there is } \lambda
\in \C^*\ \hbox{with}\ (a_0,\ldots,a_n) =
(\lambda^{q_0}b_0,\ldots,\lambda^{q_n}b_n).
$$
Obviously $\P(1,\ldots,1) = \P^n$.  We will eventually show that
$\P(q_0,\ldots,q_n)$ is a toric variety.  The following exercise shows
that $\P(q_0,\ldots,q_n)$ contains a copy of $(\C^*)^n$.
\bigskip

\ex{1.12} As above, let $q_0,\ldots,q_n$ be positive integers with
${\rm gcd}(q_0,\ldots,q_n) = 1$. 
\item{a.} Prove that $(\C^*)^{n+1}/\widetilde H \subset
\P(q_0,\ldots,q_n)$, where $\widetilde H = \{(\lambda^{q_0},\ldots,
\lambda^{q_n}) \mid \lambda \in \C^*\}$.  
\item{b.} Prove that $(\C^*)^{n+1}/\widetilde H \simeq (\C^*)^n$.
Hint:~Make $(q_0,\ldots,q_n)$ the first column of a matrix $M \in {\rm
GL}(n+1,\Z)$ and use $M$ to define an automorphism of $(\C^*)^{n+1}$.
\bigskip

We call $q_0,\ldots,q_n$ the {\it weights\/} of the weighted
projective space.  In terms of the polynomial ring
$\C[x_0,\ldots,x_n]$, this means that $x_i$ has degree $q_i$, and $f
\in \C[x_0,\ldots,x_n]$ is {\it weighted homogeneous\/} of (weighted)
degree $d$ if
$$
f(\lambda^{q_0} x_0,\ldots, \lambda^{q_n} x_n) = \lambda^d
f(x_0,\ldots,x_n). \leqno(1.2) 
$$
It is then easy to see that we can define {\it weighted projective
subvarieties\/} in $\P(q_0,\ldots,q_n)$ using weighted homogeneous
polynomials. 

There are several ways to think about weighted projective spaces.  The
following two exercises give two ways to represent $\P(1,1,2)$.
\bigskip

\ex{1.13} Consider $\P(1,1,2)$ with variables $x_0,x_1,x_2$ of degrees
$1,1,2$ repsectively.
\item{a.} Show that $x_0^2,x_0x_1,x_1^2,x_2$ are (weighted)
homogeneous of degree 2.
\item{b.} Show that $(a_0,a_1,a_2) \mapsto (a_0^2,a_0a_1,a_1^2,a_2)$
is a well-defined map $F : \P(1,1,2) \to \P^3$.
\item{c.} Show that the map $F$ of part b is injective and that its
image is the surface defined by the equation $y_0y_2-y_1^2 = 0$ (where
$y_0,y_1,y_2,y_3$ are the coordinates of $\P^3$).
\bigskip

\ex{1.14} Show that $(a_0,b_0,c_0) \to (a_0,b_0,c_0^2)$ gives a
well-defined map $\P^2 \to \P(1,1,2)$.  Also show that this map is
surjective and is two-to-one except above $(0,0,1) \in \P(1,1,2)$.
\bigskip

We can also cover a weighted projective space by affine open subsets,
though in this case, the open sets will be affine toric varieties
instead of affine space $\C^n$.  Rather than work this out in general,
we will restrict to the case of $\P(1,1,2)$.  Here, we have the
Zariski open sets $U_i = \{(a_0,a_1,a_2) \in \P(1,1,2) \mid a_i
\ne 0\}$.
\bigskip

\ex{1.15} Let $U_0,U_1,U_2$ be the subsets of $\P(1,1,2)$ defined
above.
\item{a.} Show that $U_0 \simeq \C^2$ via $(a,b,c) \mapsto
(b/a,c/a^2)$ and $U_1 \simeq \C^2$ via $(a,b,c) \mapsto (a/b,c/b^2)$.
\item{b.} Let $V = {\bf V}(xz-y^2) \subset \C^3$.  Show that $U_2
\simeq V$ via $(a,b,c) \mapsto (a^2/c,ab/c,b^2/c)$.
\item{c.} Show that $V$ is isomorphic to the affine toric variety
$U_\sigma$ for the cone $\sigma = {\rm Cone}(e_1,e_1+2e_2)$.
Hint:~Show that $S_\sigma\cap M$ is generated by $u_0 = e_2^*$, $u_1 =
e_1^*$ and $u_2 = 2e_1^*-e_2^*$.  Note that $u_0+u_2 = 2u_1$.
\bigskip

In Lecture 4, we will show that $\P(1,1,2)$ is the abstract variety
obtained by ``gluing'' two copies of $\C^2$ together with the affine
toric variety $U_\sigma$ from part c of Exercise~1.15.  But before we
can do this, we must first understand what ``gluing'' means.
\bigskip
\medskip
\goodbreak

\noindent {\bigbf \S2. Abstract Varieties}
\bigskip
\smallskip

\noindent \ul{T\sc{he} D\sc{efinition of} M\sc{anifold}}. To better
understand the definition of abstract variety, we begin by recalling
the definition of a $C^\infty$ $n$-manifold.  Such a manifold consists
of a second countable Hausdorf topological space $M$ together with an
open cover $U_\a$ and homeomorphisms $f_\a : U_\a \to V_\a$, where
$V_\a \subset \R^n$ is open, such that for every $\a,\beta$, the
composition
$$
f_\beta\circ f_\a^{-1} : f_a(U_\a\cap U_\beta) \to f_\beta(U_\a\cap
U_\beta)
$$
is a diffeomorphism.

It turns out that there is a simpler, though more sophisticated, way
of giving this definition.  We begin with an open set $V \subset
\R^n$.  The {\it sheaf\/} of $C^\infty$ functions on $V$, denoted
$\cO^\infty_V$, is defined by assigning to each open set $U \subset V$
the $\R$-algebra
$$
\cO_V^\infty(U) = \{f : U \to \R \mid f\ \hbox{is a}\ C^\infty\
\hbox{function}\}.\leqno(2.1)
$$

More generally, given a topological space $X$, we say that ${\cal F}$
is {\it sheaf of $\R$-algebras\/} on $X$ if for every open set $U
\subset X$, there is an $\R$-algebra ${\cal F}(U)$ such that:
\item{$\bullet$} If $V \subset U$ are open, then there is an  
$\R$-algebra homomorphism $r^U_{V} :{\cal F}(U) \to {\cal F}(V)$.*
\item{$\bullet$} $r^U_{U}$ is the identity and if $W \subset V \subset
U$ are open, then $r^V_{W}\circ r^U_{V} = r^U_{W}$.\footnote*{These
two bullets say that ${\cal F} : \hbox{\bf Open sets of $X$} \to
\R\hbox{\bf -Algebras}$ is a contravariant functor.}
\item{$\bullet$} If $U = \bigcup_\a U_\a$, where $U_\a$ is open, then
we have an exact sequence
$$
0 \to {\cal F}(U) \to\ \prod_\a {\cal F}(U_\a)\
\lower3pt\hbox{$\smash{\mathop{\to}\limits^{\displaystyle \to}}$}\
\prod_{\a,\beta} {\cal F}(U_\a\cap U_\beta),
$$
where the second arrow is the map ${\cal F}(U) \to {\cal F}(U_\a)$ and
the double arrows are the maps ${\cal F}(U_\a) \to {\cal F}(U_\a\cap
U_\beta)$ and ${\cal F}(U_\beta) \to {\cal F}(U_\a\cap U_\beta)$.
\vskip0pt
\noindent Elements of ${\cal F}(U)$ are called {\it sections\/} of
${\cal F}$ over $U$, and when $V \subset U$, $r^U_V : {\cal F}(U) \to
{\cal F}(V)$ the {\it restriction map\/}.  In the third bullet,
exactness at ${\cal F}(U)$ means that sections of ${\cal F}(U)$ are
determined locally, i.e., two sections are equal if and only if their
restrictions to the $U_\a$ are equal.  Exactness at $\prod_\a {\cal
F}(U_\a)$ means that compatible sections patch, i.e., sections over
the $U_\a$ which agree on their intersections come from a section over
$U$.

Given a sheaf ${\cal F}$ and $U \subset X$ open, the sections of
${\cal F}$ over $U$ can be denoted
$$
{\cal F}(U) = \Gamma(U,{\cal F}) = H^0(U,{\cal F}).
$$
We will use the first of these in this lecture but will switch to the
second for Lecture 9.  When discussing sheaf cohomology in later
lectures, we will use the third.

When ${\cal F}$ is a sheaf of $\R$-algebras on $X$, we call the pair
$(X,{\cal F})$ a {\it ringed space over\/} $\R$.  For example, when $V
\subset \R^n$ is open, (2.1) gives a ringed space over $\R$ denoted
$(V,\cO_V^\infty)$ .
\bigskip

\ex{2.1} Complete the following definitions:
\item{a.} The {\it restriction\/} ${\cal F}{\big|}_U$ of a sheaf
${\cal F}$ on $X$ to an open set $U \subset X$ is defined by \dots
\item{b.} Ringed spaces $(X,{\cal F})$, $(Y,{\cal G})$ over $\R$ are
{\it isomorphic\/} if there are a homeomorphism $\phi : X \to Y$ and,
for $U \subset Y$ open, an $\R$-algebra isomorphism $\phi^\#_U: {\cal
G}(U) \to {\cal F}(\phi^{-1}(U))$, such that \dots
\bigskip

We can now reformulate the definition of $C^\infty$ $n$-manifold.
\bigskip

\ex{2.2} Let $n$ be a positive integer and let $(M,\cO_M)$ be a ringed
space over $\R$.  Assume that every point in $M$ has a neighborhood
$U$ such that $(U,\cO_M{\big|}_U)$ is isomorphic to $(V,\cO^\infty_V)$
for some open subset $V \subset \R^n$.  Prove that $M$ has the
structure of a $C^\infty$ $n$-manifold.
\bigskip

\ex{2.3} Conversely, let $M$ be a $C^\infty$ $n$-manifold as defined
at the beginning of the section.   
\item{a.} Given $U \subset M$ open, define what it means for
$f : U \to \R$ to be $C^\infty$.  
\item{b.} Use the definition given in part a to define the
sheaf $\cO_M$ of $C^\infty$ functions on $M$ and show that $(M,\cO_M)$
is a ringed space over $\R$ which satisfies the condition of
Exercise~2.2.
\bigskip

For a $C^\infty$ $n$-manifold $M$, the sheaf $\cO_M$ of Exercise~2.3
is called the {\it structure sheaf\/} of $M$.
\bigskip

\ex{2.4} Give sheaf-theoretic definitions of a $C^k$ $n$-manifold and
a complex $n$-manifold. 
\bigskip
\smallskip

\noindent \ul{T\sc{he} S\sc{tructure} S\sc{heaf of an} A\sc{ffine}
V\sc{ariety}}.  We first show that some of the constructions for
$\C^n$ given in Lecture~1 generalize to an arbitrary affine variety $V
= {\rm Spec}(R)$.
\bigskip

\ex{2.5} Let $V = {\rm Spec}(R)$ be an affine variety.
\item{a.} Given an ideal $I \subset R$, define ${\bf V}(I) \subset V$.
Then prove that ${\bf V}(I)$ is a subvariety of $V$ and that all
subvarieties of $V$ arise in this way.
\item{b.} Given a subvariety $W \subset V$, define ${\bf I}(W) \subset
R$ and prove that ${\bf I}(W)$ is a radical ideal of $R$.
\item{c.} Prove the Nullstellensatz, i.e., that ${\bf I}({\bf V}(I))
= \sqrt{I}$ for any ideal $I \subset R$.
\item{d.} Prove the Hilbert Basis Theorem, i.e., that any ideal
$I\subset R$ can be written in the form $I = \langle
f_1,\ldots,f_s\rangle$, where $f_1,\ldots,f_s \in R$. 
\bigskip

In Lecture 1, we defined the Zariski open $V_f \subset V$ for any $f
\in R$.  Furthermore, when $V$ is irreducible, we showed that $V_f =
{\rm Spec}(R_f)$, where  
$$
R_f = \{g/f^m \in \C(V) \mid g \in R,\ m \ge 0\}
$$
is the localization of $R$ at $f$, as defined in Exercise 1.9 of
Lecture 1.
\bigskip

\ex{2.6} Show that the sets $V_f$ form a basis of the Zariski topology
of $V$.
\bigskip

The {\it structure sheaf\/} of an irreducible affine variety $V = {\rm
Spec}(R)$ is the sheaf of $\C$-algebras in the Zariski topology
defined as follows.  Given a Zariski open $U \subset V$, a function
$\phi : U \to \C$ is {\it regular\/} if for every $p \in V$, there is
$f_p \in R$ such that $p \in V_{f_p} \subset U$ and
$\phi{\big|}_{V_{f_p}} \in R_{f_p}$.  Then
$$
\cO_V(U) = \{\phi : U \to \C \mid \phi\ \hbox{is a regular function}\}.
$$
We will not show that $\cO_V$ is a sheaf of $\C$-algebras---we refer
the reader to [2] or [3] for the details of the proof.  These
references also show how to define $\cO_V$ when $V$ is not
irreducible.
\bigskip

\ex{2.7} Let $V = \C^2$ and set $U = \C^2-\{(0,0)\}$.  Show that
$\cO_V(U) = \C[x,y]$.  
\bigskip

The structure sheaf $\cO_V$ has two important properties.

\proclaim Theorem 2.1. Let $V = {\rm Spec}(R)$ be an irreducible
affine variety. 
\item{a.} $\cO_V(V) = R$.
\item{b.} If $f \in R$, then $\cO_V{\big|}_{V_f} = \cO_{V_f}$.

\prf 
For part a, it suffices to show $\cO_V(V) \subset R$.  If $\phi : V
\to \C$ is a morphism, then for each $p \in V$, there are $f_p,g_p \in
R$ such that $\phi = g_p/f_p^{m_p}$ and $f_p(p) \ne 0$.  Let $I =
\langle f_p^{m_p} \mid p \in V\rangle \subset R$.  It follows easily
that ${\bf V}(I) = \emptyset$, so that by the Nullstellensatz,
$\sqrt{I} = {\bf I}({\bf V}(I)) = {\bf I}(\emptyset) = R$.  Thus $1
\in R$, which implies that $1 = \sum_{p\in S} h_p f_p^{m_p}$ where
$h_p \in R$ and $S \subset V$ is finite.  Then
$$
\phi = \sum_{p\in S} h_p f_p^{m_p}\, \phi = \sum_{p\in S} h_p\, g_p
\in R. 
$$

For part b, let $U \subset V$ be Zariski open.  If $\phi : U
\to \C$ is a morphism, then for every $p \in U$, there is $p \in
V_{f_p} \subset U$ such that $\phi \in R_{f_p}$.  Now suppose in
addition that $U \subset V_f$.  If we regard $f_p$ as an element of
$R_f$, then one easily sees that
$$
\phi \in R_{f_p} \subset R_{ff_p} = (R_f)_{f_p}.
$$
Furthermore, $V_{f_p}\cap V_f = (V_f)_{f_p}$ shows that $p \in
(V_f)_{f_p} \subset U \subset V_f$.  By definition, this implies that
$\phi \in \cO_{V_f}(U)$, and part b now follows easily.\qed
\medskip

Combining parts a and b of Theorem 2.1, we conclude that
$$
\cO_V(V_f) = \cO_V{\big|}_{V_f}(V_f) = \cO_{V_f}(V_f) = R_f
$$
when $V = {\rm Spec}(R)$ and $f \in R$.

\smallskip
\bigskip
\noindent \ul{T\sc{he} D\sc{efinition of} A\sc{bstract}
V\sc{ariety}}.  We now give the main definition of this
section.

\proclaim Definition 2.2. An {\bf abstract variety\/} $(X,\cO_X)$ is a
ringed space over $\C$ where each $p \in X$ has a neighborhood $U$
such that the restriction $(U,\cO_X{\big|}_U)$ is isomorphic $($as a
ringed space over $\C)$ to $(V,\cO_V)$ for some affine variety $V$.

Given an abstract variety $(X,\cO_X)$, an open set $U \subset X$ is an
{\it affine open\/} if $(U,\cO_X{\big|}_U)$ is isomorphic (over $\C$)
to the ringed space of an affine variety.  The topology on $X$ is
called the {\it Zariski topology\/} since it restricts to the Zariski
topology in each affine open subset.
\bigskip

\ex{2.8} Let $(X,\cO_X)$ be an abstract variety and let $U \subset X$
be Zariski open.  Show that every section $\phi \in \cO_X(U)$ gives a
function $\phi : U \to \C$.  We say that $\phi$ a {\it regular
function\/} on $U$.
\bigskip

\ex{2.9} Let $(X,\cO_X)$ be an abstract variety and let $Y \subset X$
be Zariski closed.  If $U_1 \subset Y$ is open, define $\cO_Y(U_1)$ to
be the set of all functions $\phi : U_1 \to \C$ such that for every $p
\in U_1$, there is $U \subset X$ open and a regular function
$\tilde\phi : U \to \C$ with $p \in U\cap Y \subset U_1$ and
$\phi{\big|}_{U\cap Y} = \tilde\phi{\big|}_{U\cap Y}$.
\item{a.} Show that $U_1 \mapsto \cO_Y(U_1)$ is a sheaf of
$\C$-algebras on $Y$.
\item{b.} When $(X,\cO_X)$ is an affine variety and $Y \subset X$ is a
subvariety, prove that the sheaf defined in part a is precisely the
sheaf of regular functions on $Y$
\item{c.} When $(X,\cO_X)$ is an abstract variety and $Y \subset X$ is
Zariski closed, prove that the sheaf of part a makes $(Y,\cO_Y)$ into
an abstract variety.
\bigskip

Given an abstract variety $(X,\cO_X)$, we say that $Y \subset X$ is a
{\it subvariety\/} if it is Zariski closed.  The above exercise shows
that $Y$ inherits the structure of an abstract variety in a natural
way.  We define $(X,\cO_X)$ to be {\it irreducible\/} if $X$ is not
the union of two proper subvarieties.
\bigskip

\ex{2.10} Prove that an abstract variety is irreducible if and only if
it is connected and every affine open subset is irreducible. 
\bigskip 

Let us show that $\P^n$ can be regarded as an abstract variety.  In
order to define the structure sheaf $\cO_{\P^n}$, we will use the
field of rational functions $\C(\P^n)$ defined in Exercise~1.6 of this
lecture.  If $U \subset \P^n$ is Zariski open, then a function $\phi :
U \to \C$ is {\it regular\/} if for each $p \in U$, there is $f/g \in
\C(\P^n)$ such that $g(p) \ne 0$ and $\phi{\big|}_{U\cap {\bf V}(g)} =
(f/g){\big|}_{U\cap {\bf V}(g)}$.  Then 
$$
\cO_{\P^n}(U) = \{ \phi : U \to \C \mid \phi\ \hbox{is a regular
function}\}
$$ 
defines a sheaf $\cO_{\P^n}$ on $\P^n$.  We also have the affine open
sets $U_i = \{(a_0,\ldots,a_n) \in \P^n \mid a_i \ne 0\}$.  In
Exercise~1.11, we noted that if we regard
$x_0/x_i,\ldots,x_{i-1}/x_i,x_{i-1}/x_i,\ldots,x_n/x_i$ as coordinates
on $\C^n$, then the map $U_i \simeq \C^n$ induces an isomorphism
$$
\C(\P^n) \simeq
\C(x_0/x_i,\ldots,x_{i-1}/x_i,x_{i-1}/x_i,\ldots,x_n/x_i).
$$
Using this isomorphism, it is easy to see that
$(U_i,\cO_{\P^n}{\big|}_{U_i})$ is isomorphic to $(\C^n,\cO_{\C^n})$.
This proves that $\P^n$ is an abstract variety.  By Exercise~2.10, we
see that $\P^n$ is irreducible.

As is customary, we often write an abstract variety $(X,\cO_X)$ as
simply $X$, and we will also drop the ``abstract''.  Thus, ``the
variety $X$'' is short for ``the abstract variety $(X,\cO_X)$''.

Finally, a variety $X$ also has a {\it classical topology\/}, which is
the coarsest topology on $X$ that agrees with the classical topology
on every affine open subset of $X$.  The structure sheaf $\cO_X$ is
not a sheaf in the classical topology.  However, one can define the
closely related sheaf $\cO_X^{\rm an}$ of {\it analytic functions\/}
on $X$, which is a sheaf in the classical topology.  We call
$(X,\cO_X^{\rm an})$ the {\it complex analytic space\/} associated to
the variety $X$.  See [3, Appendix B] and [4] for more details.

\smallskip
\bigskip
\noindent \ul{T\sc{he} F\sc{unction} F\sc{ield of an}
I\sc{rreducible} V\sc{ariety}}.  If $X$ is irreducible, then a {\it
rational function\/} on $X$ is a regular function $\phi : U \to \C$,
where $U$ is a nonempty Zariski open.  Two rational functions are {\it
equivalent\/} if they agree on some nonempty Zariski open, and the set
of equivalence classes is denoted $\C(X)$.  One can prove that $\C(X)$
is a field, called the {\it function field\/} of $X$.
\bigskip

\ex{2.11} Let $U$ be an affine open of an irreducible variety
$X$.  Prove that $\C(U) \simeq \C(X)$.
\bigskip

We say that $\phi \in \C(X)$ is {\it defined\/} at $p \in X$ there is
a regular function $\phi' : U \to \C$ such that $p \in U$ and $\phi$
is equivalent to $\phi'$.  For $\phi \in \C(X)$, the set $\{p \in X
\mid \phi\ \hbox{is defined at}\ p\}$ is the largest Zariski open on
which $\phi$ is defined.

\smallskip
\bigskip
\noindent \ul{T\sc{he} L\sc{ocal} R\sc{ing of a} P\sc{oint}}.  Given
an irreducible variety $X$ and a point $p \in X$, we define 
the {\it local ring\/} of $X$ at $p$ to be
$$
\cO_{X,p} = \{\phi \in \C(X) \mid \phi\ \hbox{is defined at}\ p\}.
$$
The key feature of $\cO_{X,p}$ is described in the following exercise.
\bigskip

\ex{2.12} Let $\cO_{X,p}$ defined as above.
\item{a.} Show that $\frakm_{X,p} = \{\phi \in \cO_{X,p} \mid \phi(p)
= 0\}$ is a maximal ideal of $\cO_{X,p}$.
\item{b.} Given $\phi \in \cO_{X,p}$, show that $\phi(p) \ne 0$
implies that $\phi^{-1} \in \cO_{X,p}$.
\item{c.} Use part b to show that $\frakm_{X,p}$ is the unique maximal 
ideal of $\cO_{X,p}$.
\bigskip 

In general, a commutative ring $R$ with unit is called a {\it local
ring\/} if it has a unique maximal ideal $\frakm$.  Thus $\cO_{X,p}$
is a local ring by Exercise~2.12.

For an arbitrary variety $X$ (not necessarily irreducible), one can
define the local ring $\cO_{X,p}$ to be the direct limit
$$
\cO_{X,p} = \lim_{{\scriptstyle \longrightarrow} \atop {\scriptstyle
p\in U}} \cO_X(U), 
$$
where the limit is over all neighborhoods $U$ of $p$.  This is
described in [2] and [3].  

\smallskip
\bigskip
\noindent \ul{M\sc{orphisms}}.  A {\it morphism\/} or {\it regular
map\/} consists of a continuous map $\phi : X \to Y$ and, for each
Zariski open $U \subset Y$, a $\C$-algebra homomorphism $\phi^\# :
\cO_Y(U) \to
\cO_X(\phi^{-1}(U))$, such that:
\item{$\bullet$} $\phi^\#$ is compatible with restriction maps.
\item{$\bullet$} For each $p \in X$, the map of local rings $\phi^\#_p :
\cO_{Y,\phi(p)} \to \cO_{X,p}$ induced by $\phi^\#$ is a {\it local
homomorphism}, meaning that $\frakm_{Y,\phi(p)} =
(\phi^\#_p)^{-1}(\frakm_{X,p})$. 
\vskip0pt
\noindent A morphism $(\phi,\phi^\#) : (X,\cO_X) \to (Y,\cO_Y)$ is
usually written $\phi : X \to Y$.

An important results is that if $V = {\rm Spec}(R)$ and $W = {\rm
Spec}(S)$ are irreducible affine varieties, then giving a morphism
$\phi : V \to W$ is equivalent to giving a $\C$-algebra homomorphism
$\phi^* : S \to R$.  This is proved in [2] and [3].    

\smallskip
\bigskip
\noindent \ul{G\sc{luing} T\sc{ogether} A\sc{ffine} V\sc{arieties}}.
We first observe that a variety $X$ can be constructed by ``gluing
together'' affine varieties along Zariski open subsets.  Namely,
Definition~2.2 implies that $X$ has an affine open cover $U_\a$, so
that $f_\a : U_\a \simeq V_\a$, where $V_\a$ is an affine variety.
Then, for any $\a,\beta$, the set
$$
V_{\a\beta} = f_\a(U_\a\cap U_\beta) \subset V_\a
$$
is Zariski open in $V_\a$, and the map
$$
g_{\a\beta} = f_\beta\circ f_\a^{-1} : V_{\a\beta} \to V_{\beta\a}
$$
is an isomorphism of Zariski open subsets.  Furthermore, these maps
are compatible as follows:
\smallskip
\item{$\bullet$} $g_{\a\a} = 1_{V_\a}$ for every $\a$.
\smallskip
\item{$\bullet$} $g_{\beta\gamma}{\big|}_{V_{\beta\a}\cap
V_{\beta\gamma}} \circ g_{\a\beta}{\big|}_{V_{\a\beta}\cap
V_{\a\gamma}} = g_{\a\gamma}{\big|}_{V_{\a\beta}\cap
V_{\a\gamma}}$ for every $\a,\beta,\gamma$.
\smallskip
\noindent We call these the {\it compatibility conditions\/}.

Conversely, suppose we have a collection
$$
(\{V_\a\}_\a,\{V_{\a\beta}\}_{\a,\beta},\{g_{\a\beta}\}_{\a,\beta}),  
$$
where each $V_\a$ is an affine variety, $V_{\a\beta} \subset V_\a$ is
Zariski open, and the $g_{\a\beta} : V_{\a\beta} \simeq
V_{\beta\a}$ are isomorphisms which satisfy the above compatibility
conditions.  Then we get the topological space
$$
X = \coprod_\a V_\a/\sim
$$
where $a \in V_\a$ is equivalent to $b \in V_\beta$ if $a \in
V_{\a\beta}$ and $b = g_{\a\beta}(a)$.  Furthermore, the structure
sheaves $\cO_{V_\a}$ patch to give a sheaf $\cO_X$, and from here it
is straightforward to prove that $(X,\cO_X)$ is a variety with an
affine open cover $U_\a$ such that $U_\a \simeq V_\a$ for every $\a$.
In this situation, we say that $X$ is obtained from the $V_\a$ by {\it
gluing\/} them together along the $V_{\a\beta}$ via the
$g_{\a\beta}$.
\bigskip

\ex{2.13} Let $V_0 = V_1 = \C$,  $V_{01} = V_{10} = \C-\{0\}$ and
$g_{01}(x) = g_{10}(x) = x^{-1}$.  Prove that this data determines
the variety $\P^1$.  

\smallskip
\bigskip
\noindent \ul{C\sc{artesian} P\sc{roducts}}.  If $X$ and $Y$ are
varieties, then their cartesian product $X\times Y$ exists, though the
definition is subtle.  The difficulty is that the usual product
topology on $X\times Y$ gives the wrong topology.  Here is an example.
\bigskip

\ex{2.14} Show that the product topology on $\C\times\C$, where we use
the Zariski topology on each factor, is not the Zariski topology on
$\C^2$. 
\bigskip

To get the correct definition of cartesian product, we begin with
affine varieties.  Suppose that $V = {\bf V}(f_1,\ldots,f_s) \subset
\C^n$, with variables $x_1,\ldots,x_n$ and $W = {\bf
V}(g_1,\ldots,g_t) \subset \C^m$, with variables $y_1,\ldots,y_m$.
Also let $R$ and $S$ be the coordinate rings of $V$ and $W$
respectively.
\bigskip

\ex{2.15} Let $V = {\rm Spec}(R)$ and $W = {\rm Spec}(S)$ be as above.
\item{a.} Show that $V \times W \subset \C^n\times\C^m = \C^{n+m}$ is
the affine variety ${\bf V}(f_1,\ldots,f_s,g_1,\ldots,g_t)$, where
$f_i(x_1,\ldots,x_n),g_j(y_1,\ldots,y_m) \in
\C[x_1,\ldots,x_n,y_1,\ldots,y_m]$. 
\item{b.} Show that the coordinate ring of $V\times W$ is $R\otimes_\C
S$.
\vskip0pt
\noindent Thus the cartesian product of $V = {\rm Spec}(R)$ and $W =
{\rm Spec}(S)$ is $V\times W = {\rm Spec}(R\otimes_\C S)$.
\bigskip

\ex{2.16} Suppose that $\sigma_1 \subset (N_1)_\R$ and $\sigma_2
\subset (N_2)_\R$ are strongly convex rational polyhedral cones.  Let
$N = N_1\oplus N_2$.
\item{a.} Show that $\sigma_1\times\sigma_2 \subset N_\R$ is a
strongly convex polyhedral cone.
\item{b.} Show that $S_{\sigma_1\times\sigma_2} \simeq
S_{\sigma_1}\oplus S_{\sigma_2}$ as semigroups.
\item{c.} Show that $\C[S_{\sigma_1\times\sigma_2}] \simeq
\C[S_{\sigma_1}] \otimes_\C \C[S_{\sigma_2}]$ as $\C$-algebras.
\item{d.} Conclude that the affine toric variety
$U_{\sigma_1\times\sigma_2}$ is the cartesian product
$U_{\sigma_1}\times U_{\sigma_2}$.
\bigskip

In the general case, we think of $X$ as obtained by gluing together
Zariski open subsets of $U_\a$, and similarly $Y$ comes from gluing
together Zariski open subsets of $U_\beta'$.  Then $X\times Y$ is
constructed by gluing together the affine varieties $U_\a \times
U_\beta'$ along suitable Zariski open subsets.  We omit the details of
the construction, which can be found in [2] and [3].

As an example, $\P^n\times\P^m$ can be constructed by this method.  If
$x_0,\ldots,x_n$ are coordinates on $\P^n$ and $y_0,\ldots,y_m$ are
coordinates on $\P^m$, then one can show that $V \subset
\P^n\times\P^m$ is Zariski closed if and only if $V = {\bf
V}(f_1,\ldots,f_s)$, where $f_i \in \C[x_0,\ldots,x_n;y_0,\ldots,y_m]$
is {\it bihomogeneous\/}, meaning that it is separately homogeneous in
the $x_i$ and in the $y_j$.

Finally, we should mention that if $X$ and $Y$ are varieties, then the
classical topology on $X\times Y$ is the product of the classical
topologies on $X$ and $Y$ respectively.
\bigskip
\medskip
\goodbreak

\noindent {\bigbf \S3. Separated, Quasi-Compact, Complete, and Normal
Varieties} 
\bigskip
\smallskip

\noindent \ul{S\sc{eparatedness}}. Given any variety $X$, the {\it
diagonal map\/} of $X$ is the map $\Delta : X \to X\times X$ defined
by $\Delta(p) = (p,p)$ for $p \in X$.  Then $X$ is {\it separated\/}
if the image of the diagonal map $\Delta(X)$ is Zariski closed in
$X\times X$.  Here are some examples of separated and non-separated
varieties.
\bigskip

\ex{3.1} Consider the variety $X$ constructed by identifying two
copies of $\C$ along $\C-\{0\}$ (in the language of \S2, this
corresponds to $V_0 = V_1 = \C$, $V_{01} = V_{10} = \C-\{0\}$ and
$g_{01}(x) = g_{10}(x) = x$).  Show that $X$ is not separated.
\bigskip

\ex{3.2} Prove that $\C^n$ is separated by considering ${\bf V}(x_1 -
y_1,\ldots,x_n-y_n) \subset \C^n\times\C^n$.
\bigskip

\ex{3.3} Prove that a subvariety of a separated variety is separated. 
\bigskip

Combining Exercises 3.2 and 3.3, we see that affine varieties are
always separated.  In Lecture 4, we will see that $\P^n$ is separated
since it is a toric variety.  By Exercise~3.3, it follows that every
projective variety is separated.

In [4], Serre characterized separatedness in terms of the classical
topology as follows.

\proclaim Theorem 3.1.  A variety $X$ is separated if and only if it
is Hausdorff in the classical topology.  

For example, this theorem makes it easy to see that the variety $X$ of
Exercise~3.1 is not Hausdorff (the two copies of the origin in $X$ do
not have disjoint neighborhoods).  This also gives a quick proof of
Exercise~3.2 since $\C^n$ is Hausdorff in the classical topology.

Here are some properties of separated varieties.

\proclaim Proposition 3.2.  Let $X$ be a separated variety.
\item{a.} If $U$ and $V$ are affine open subsets of $X$, then $U\cap
V$ is also an affine open of $X$.
\item{b.} If $f,g : Y \to X$ is a morphism of varieties, then
$\{y \in Y \mid f(y) = g(y)\}$ is a subvariety of $Y$.

\prf
For part a, we note that $\Delta : X \to X\times X$ identifies $U\cap
V$ with $\Delta(X)\cap (U\times V)$.  We know that $\Delta(X)$ is
Zariski closed in $X\times X$ by the definition of separated.  It
follows that $\Delta(X)\cap (U\times V)$ is Zariski closed in $U\times
V$.  But $U\times V$ is affine by Exercise~2.15, which implies that
$\Delta(X)\cap (U\times V) \simeq U\cap V$ is also affine.

To prove part b, consider $\phi : Y \to X\times X$ defined by $\phi(y)
= (f(y),g(y))$.  Since
$$
\{y \in Y \mid f(y) = g(y)\} = \phi^{-1}(\Delta(X)),
$$
we see that $\{y \in Y \mid f(y) = g(y)\}$ is Zariski closed in $Y$
since $\phi$ is continuous in the Zariski topology and (by
separatedness) $\Delta(X)$ is Zariski closed in $X\times X$.\qed
\medskip 

When studying differentiable manifolds, one always assumes that the
underlying topological space is Hausdorff.  Similarly, in algebraic
geometry, the varieties of interest are almost always separated.  For
this reason, we henceforth reserve the term {\it variety\/} for a
separated variety.  A non-separated variety will be called a {\it
pre-variety\/}.  The toric varieties defined in the next lecture will
be all separated (though there have been some recent papers on toric
pre-varieties).

\smallskip
\bigskip
\noindent \ul{Quasi-Compactness}. We say that a variety $X$ is {\it
quasi-compact} if $X$ is the union of finitely many affine open
subsets.  Any affine variety is quasi-compact.  An more interesting
example is $\P^n$, which is quasi-compact since it the union of the 
affine open subsets $U_i = \P^n-{\bf V}(x_i)$.

Here are the main properties of quasi-compact varieties.

\proclaim Proposition 3.3. Let $X$ be a quasi-compact variety.  Then:
\item{a.} Every subvariety of $X$ is quasi-compact.
\item{b.} Every Zariski open subset of $X$ is quasi-compact.
\item{c.} Every Zariski open cover of $X$ has a finite subcover.

\prf
Suppose that $X = U_1\cup\cdots\cup U_r$ where $U_i$ is an affine open
subset of $X$.  Then part a is obvious since $Y = (Y\cap
U_1)\cup\cdots\cup (Y\cap U_r)$ where $Y\cap U_i$ is an affine open of
$Y$.

For part b, $U = (U\cap U_1)\cup\cdots\cup (U\cap U_r)$ shows that we
can assume that $X = {\rm Spec}(R)$ is affine.  Then $X-U = {\rm
V}(I)$ for some ideal $I \subset R$.  The Hilbert basis theorem
implies $I = \langle f_1,\ldots,f_s\rangle$, and it follows that $U =
\bigcup_{i=1}^s X_{f_i}$.

Finally, we leave part c as Exercise 3.4 below.\qed
\bigskip

\ex{3.4} Prove that every Zariski open cover of a quasi-compact
variety has a finite subcover. 
\bigskip

This exercise explains there the term ``quasi-compact'' comes from.
As with separatedness, the varieties of interest to algebraic
geometers are almost always quasi-compact.  Hence, from now on,
whenever we say {\it variety\/}, we will mean a separated
quasi-compact abstract variety.  The toric varieties defined in
Lecture 4 will be quasi-compact (though one sometimes considers toric
varieties coming from infinite fans, which are not quasi-compact).

\smallskip
\bigskip
\noindent \ul{C\sc{ompleteness and} P\sc{roperness}}. A variety $X$ is
{\it complete} if the following conditions hold:
\item{$\bullet$} $X$ is separated.
\item{$\bullet$} $X$ is quasi-compact.
\item{$\bullet$} For every other variety $Y$, the projection map
$X\times Y \to Y$ is closed, meaning that the projection of a Zariski
closed set in $X\times Y$ is Zariski closed in $Y$.
\bigskip

\ex{3.5} Prove that a subvariety of a complete variety is complete.
\bigskip

\ex{3.6} Prove that the cartesian product of two complete varieties is
complete. 
\bigskip

\ex{3.7} Prove that a variety $X$ is complete if and only if for all
$m \ge 1$, the projection map $X \times \C^m \to \C^m$ is closed.
\bigskip

The most basic example of a complete variety is $\P^n$.  

\proclaim Theorem 3.4. $\P^n$ is complete.

We will prove this in Lecture 6 when we consider the toric variety of
a complete fan.  For another proof, note that Theorem 6 of [1,
Chapter~8, \S5] implies that the projection map $\P^n\times\C^m \to
\C^m$ is closed.  By Exercise~3.7, it follows that $\P^n$ is
complete, and then any projective variety is complete by Exercise~3.5.

The completeness of $\P^n$ is closely related to elimination theory.
To see why, suppose we have polynomials
$$
f_1,\ldots,f_s \in \C[x_0,\ldots,x_n,y_1,\ldots,y_m]
$$
which are homogeneous in $x_0,\ldots,x_n$.  We can think of the $f_i$
as homogeneneous polynomials in the $x_i$ whose coefficients depend on
the ``parameters'' $y_j$.  Question:~For which values of the parameters
$y_j$ do the equations
$$
f_1 = \cdots = f_s = 0\leqno(3.1)
$$
have a nontrivial solution in the $x_i$?

To answer this question, observe that (3.1) defines a subvariety $W
\subset \P^n\times\C^m$, and the values of the $y_j$ for which (3.1)
has a nontrivial solution is the image of $W$ under the projection map
$\P^n\times\C^m \to \C^m$.  Since $\P^n$ is complete, this image is a
variety in $\C^m$.  In other words, there are polynomials
$g_1,\ldots,g_l \in \C[y_1,\ldots,y_m]$ such that (3.1) has a
nontrivial solution for the parameter values $y_j = b_j$ if and only
if
$$
g_1(b_1,\ldots,b_m) = \cdots = g_l(b_1,\ldots,b_m) = 0.\leqno(3.2)
$$
Chapter 8 of [1] gives an algorithm for finding the polynomials $g_i$.
We say that (3.2) is obtained from (3.1) by {\it eliminating\/} the
variables $x_0,\ldots,x_n$.  This is {\it projective elimination
theory\/}.

Serre's paper [4] characterizes completeness in terms of the classical
topology as follows.

\proclaim Theorem 3.5. A variety $X$ is complete if and only if it is
compact in the classical topology.

This theorem and the Hopf fibration $S^{2n+1} \to \P^n$ give another
proof that $\P^n$ is complete (see Exercise~1.2).  In algebraic
geometry, completeness is a very useful property.  Here is a result
which indicates some reasons why.

\proclaim Theorem 3.6. Let $X$ be a complete variety.
\item{a.} If $\phi : X \to Y$ is a morphism, then its image $\phi(X)$
is a subvariety of $Y$.
\item{b.} If $\phi : \C^* \to X$ is a morphism, then $\phi$ extends
uniquely to a morphism $\tilde\phi : \C \to X$.
\item{c.} $X$ is affine if and only if $X$ is a finite set of points.
\item{d.} If $X$ is connected, then every morphism $\phi : X \to \C$
is constant.

Part b of this theorem says that if $X$ is complete and $\phi : \C^*
\to X$ is a morphsim, then $\lim_{t\to 0} \phi(t)$ exists as a unique
element of $X$ whenever $X$ is complete (in fact, $\lim_{t\to \infty}
\phi(t)$ also exists, so that $\phi$ extends to a map $\P^1 \to X$).
Part c says that affine varieties are very far from being complete
since these concepts coincide only for finite sets of points.

Finally, completeness closely related to the idea of a {\it proper
morphism\/} $\phi : X \to Y$.  We will not give the definition here
(we would need to define {\it fibered products\/} and {\it morphisms
of finite type\/}).  The reader should consult [2] and [3] for the
full definition.  Turning to Serre's classic paper [4] yet again, we
can characterize properness in terms of the classical topology as
follows.

\proclaim Theorem 3.7. A morphism $\phi : X \to Y$ is proper if and
only if it is proper in the classical topology, meaning that
$\phi^{-1}(C) \subset X$ is compact whenever $C \subset Y$ is compact.
\bigskip

\ex{3.8} Prove that $X$ is complete if and only if $X \to \{{\rm
pt}\}$ is proper, where $\{{\rm pt}\}$ is the variety consisting of a
single point.

\smallskip
\bigskip
\noindent \ul{N\sc{ormality}}.  A variety $X$ is {\it normal\/} if it
is irreducible and the local ring $\cO_{X,p}$ is integrally closed for
every $p \in X$.  In order to relate this to the definition of normal
affine variety given in Lecture~1, we first generalize Exercise~3.2 of
Lecture~2.
\bigskip

\ex{3.9} Let $R$ be an integral domain with field of fractions $K$.
A subset $S \subset R$ is a {\it multiplicative subset\/} if $1 \in
S$, $0 \notin S$, and $S$ is closed under multiplication.  Then the
{\it localization\/} of $R$ at $S$ is $R_S = \{a/b \in K \mid a \in R,
b \in S\}$.
\item{a.} Prove that $R_S$ is the smallest subring of $K$ containing
$R$ such that every $s \in S$ is invertible in $R_S$.
\item{b.} Prove that if $R$ is integrally closed, then $R_S$ is
integrally closed.
\bigskip

We now show that for affine varieties, our two notions of normal
coincide. 

\proclaim Proposition 3.8.  Let $V = {\rm Spec}(R)$ be an irreducible
affine variety.  The $R$ is integrally closed if and only if the local
ring $\cO_{V,p}$ is integrally closed for all $p \in V$.

\prf
We know from Lecture 1 that $p \in V$ corresponds to a maximal ideal
$M \subset R$.  Then $R-M$ is a multiplicative subset since maximal
implies prime, and one easily shows that
$$
\cO_{V,p} = R_{R-M}.
$$
If $R$ is integrally closed, then Exercise 3.9 implies that
$\cO_{V,p}$ is also integrally closed.  Conversely, if all of the
$\cO_{V,p}$ are integrally closed, then so is
$$
\bigcap_{p\in V} \cO_{V,p}
$$
by Exercise 3.3 of Lecture 2.  However, this intersection is precisely
$\cO_V(V)$, which equals $R$ by part a of Theorem 2.1 of this lecture.
It follows that $R$ is integrally closed.\qed
\medskip

Proposition 3.8 has a following immediate corollary.

\proclaim Corollary 3.9. An irreducible variety $X$ is normal
if and only if it is a union of affine varieties $U_\a = {\rm
Spec}(R_\a)$ where $R_\a$ is integrally closed.

\bigskip
\noindent {\bigbf \S4. Gluing Affine Toric Varieties}
\bigskip

In \S2 we learned how to construct varieties by gluing together
Zariski open subsets of affine varieties.  For affine toric varieties,
we can use the faces of a cone to specify the desired gluing.

The full details of this will be given in Lecture 4.  Here, we will
just indicate the basic ideas and give a simple example.  To begin,
suppose that $\sigma \subset N_\R$ is a strongly convex rational
polyhedral cone.  Then, if $\tau$ is a face of $\sigma$, we get affine
toric varieties $U_\sigma = {\rm Spec}(\C[S_\sigma])$ and $U_\tau =
{\rm Spec}(\C[S_\tau])$.  We first determine how these are related.

In Proposition 5.5 of Lecture 1, we showed that $\tau = H_u\cap\sigma$
for some $u \in S_\sigma$ and that
$$
S_\tau = S_\sigma + \Z_{\ge 0}\cdot(-u).
$$
Furthermore, by Proposition 2.3 of Lecture 2, this implies that the
natural map $U_\tau \to U_\sigma$ is injective and that its image can
be identified with the Zariski open subset $(U_\sigma)_{\chi^u}$.
Thus
$$
U_\tau \simeq (U_\sigma)_{\chi^u}.\leqno(4.1)
$$ 

Now suppose that $\sigma$ and $\sigma'$ are strongly convex rational
polyhedral cones in $M_\R$ whose intersection $\tau =
\sigma\cap\sigma'$ is a face of each.  In Exercise~5.6 of Lecture~1,
you showed how to find $u \in M$ such that
$$
\eqalign{\tau &= H_u\cap \sigma,\ u \in S_\sigma\cr
	 \tau &= H_{-u}\cap \sigma',\ -u \in
	 S_{\sigma'}.\cr}\leqno(4.2) 
$$
Applying (4.1), we get 
$$
U_\sigma \supset (U_\sigma)_{\chi^u} \simeq U_\tau \simeq
(U_{\sigma'})_{(\chi^{u})^{-1}} \subset U_{\sigma'}.\leqno(4.3)
$$
As you can see, this is the perfect set-up for gluing $U_\sigma$ and
$U_{\sigma'}$ in such a way that $U_\tau$ becomes their intersection.
Here is a simple example of what this looks like in practice.
\bigskip

\ex{4.3} Let $N = \Z$ and set $\sigma = {\rm Cone}(e_1)$, $\sigma' =
{\rm Cone}(-e_1)$.  Note that $\tau = \{0\}$.
\item{a.} Show that we can pick $u = e_1$ in (4.2).
\item{b.} Show that (4.3) can be written as
$$
\C = {\rm Spec}(\C[x]) \supset {\rm Spec}(\C[x])_x = \C^* = {\rm
Spec}(\C[x^{-1}])_{x^{-1}} \subset {\rm Spec}(\C[x^{-1}]) = \C.
$$
\item{c.} Show that this data is equivalent to the data of
Exercise~2.13 of this lecture.
\item{d.} Conclude that $\P^1$ is the variety obtained by gluing
$U_\sigma$ and $U_{\sigma'}$ together along $U_\tau$.
\bigskip

This exercise shows that we can give a purely ``toric'' construction
of $\P^1$.  Lecture~4 will use a combinatorial device called a {\it
fan\/} to construct arbitrary toric varieties in the same way.

\bigskip
\noindent{\bigbf References}
\bigskip

\parindent=12pt

\ref 1.~D.\ Cox, J.\ Little and D.\ O'Shea, {\sl Ideals, Varieties,
and Algorithms\/}, Second Edition, Springer-Verlag, New
York-Berlin-Heidelberg, 1997.
\smallskip

\ref 2.~D.\ Eisenbud and J.\ Harris, {\sl The Geometry of Schemes\/},
Springer-Verlag, New York-Berlin-Heidelberg, 2000.
\smallskip

\ref 3.~R.\ Hartshorne, {\sl Algebraic Geometry\/}, Springer-Verlag,
New York-Berlin-Heidelberg, 1977. 
\smallskip

\ref 4.~J.-P.\ Serre, {\it G\'eom\'etrie alg\'ebrique et g\'eom\'etrie
analytique\/}, Ann.\ Inst.\ Fourier {\bf 6} (1956), 1--42.

\end