% Notes on Algebraic Geometry
% February 6, 2001
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% Grenoble Toric Summer School

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to Algebraic Geometry} \strut}

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\bigskip
\bigskip


\medskip

\centerline{\bigrm David A. Cox}

\medskip

\centerline{\bigrm Department of Mathematics and Computer Science}

\medskip

\centerline{\bigrm Amherst College}

\bigskip
\medskip
\bigskip
\medskip

\centerline{\bigbf Table of Contents}
\bigskip

\noindent Acknowledgements \dotfill 2
\medskip

\noindent\S1. Affine Varieties \dotfill 2
\medskip

\noindent\S2. Projective Varieties \dotfill 6
\medskip

\noindent\S3. Abstract Varieties \dotfill 10
\medskip

\noindent\S4. Separated, Quasi-Compact, Complete and Normal Varieties
\dotfill 15
\medskip

\noindent\S5. Smooth and Quasismooth Varieties \dotfill 19
\medskip

\noindent\S6. The Local Ring of an Irreducible Hypersurface \dotfill 23
\medskip

\noindent\S7. Weil Divisors on Normal Varieties \dotfill 24
\medskip

\noindent\S8. Cartier Divisors on Normal Varieties \dotfill 27
\medskip

\noindent\S9. The Sheaf of a Weil Divisor \dotfill 28
\medskip

\noindent\S10. Invertible Sheaves and Line Bundles \dotfill 31
\medskip

\noindent References \dotfill 37

\vfill\eject

\noindent {\bigbf Acknowledgements}
\bigskip
\smallskip

These notes are intended to help students make the transition between
the elementary aspects of algebraic geometry (varieties in affine and
projective space, etc.)~and some of its more sophisticated aspects
(normal varieties, Weil and Cartier divisors, etc.).

The material presented here is taken from the notes prepared for the
course on toric varieties given at the Summer School on {\it The
Geometry of Toric Varieties\/} held in Grenoble in 2000.  I am
grateful to Laurent Bonavero and Michel Brion for inviting me to
participate in the Summer School.  Their comments (and those of
Gottfried Barthel and the other participants) are greatly appreciated.
I also thank Henning \'Ulfarsson for pointing out some typos to fix.

\bigskip
\medskip
\noindent {\bigbf \S1.  Affine Varieties}
\bigskip
\smallskip

\noindent \ul{B\sc{asic} D\sc{efinitions}}. For simplicity, we will work
over the complex numbers $\C$.  Then, given polynomials
$f_1,\ldots,f_s \in \C[x_1,\ldots,x_n]$, we get the {\it affine
variety\/}
$$
{\bf V}(f_1,\ldots,f_s) = \{a \in \C^n \mid f_1(a) = \cdots = f_s(a) =
0\}.
$$
More generally, if $I \subset \C[x_1,\ldots,x_n]$ is an ideal, then we
define 
$$
{\bf V}(I) = \{a \in \C^n \mid f(a) = 0\ \hbox{for all}\ f \in I\}.
$$
\smallskip

\ex{1.1} Let $I = \langle f_1,\dots,f_s\rangle \subset
\C[x_1,\ldots,x_n]$ be the ideal generated by $f_1,\dots,f_s$.  Show
that ${\bf V}(I) = {\bf V}(f_1,\dots,f_s)$.  (All ideals in
$\C[x_1,\ldots,x_n]$ are of this form by the Hilbert Basis Theorem.)
\bigskip

Conversely, given an affine variety $V \subset \C^n$, we get the ideal 
$$
{\bf I}(V) = \{f \in \C[x_1,\ldots,x_n] \mid f(a) = 0\ \hbox{for all}\
a \in V\}.
$$

\smallskip

\ex{1.2} Let $V \subset \C^n$ be an affine variety and $I \subset
\C[x_1,\ldots,x_n]$ an ideal.  Show that:
\item{a.} $V = {\bf V}({\bf I}(V))$.
\item{b.} $I \subset \sqrt{I} \subset {\bf I}({\bf V}(I))$, where
$\sqrt{I} = \{f \in \C[x_1,\ldots,x_n] \mid f^m \in I,\ m \ge 1\}$ is
the {\it radical\/} of $I$. 
\bigskip

The above exercise actually works over any field.  But since $\C$ is
algebraically closed, we also have the following basic result of
Hilbert.

\proclaim Hilbert Nullstellensatz. For any ideal $I \subset
\C[x_1,\ldots,x_n]$, we have $\sqrt{I} = {\bf I}({\bf V}(I))$.

A proof can be found in Chapter 4 of [3].  This theorem allows us to
translate algebra into geometry and vice versa.  Here is an example.
\bigskip

\ex{1.3} Use the Nullstellensatz to show the following.
\item{a.} Every maximal ideal of $\C[x_1,\ldots,x_n]$ is of the form
$\langle x_1-a_1,\ldots,x_n-a_n\rangle$, where $a_i \in \C$. Thus
there is a one-to-one correspondence between points of $\C^n$ and
maximal ideals of $\C[x_1,\ldots,x_n]$.
\item{b.} An ideal $I$ is {\it radical\/} if $I = \sqrt{I}$.  Show
that the correspondence of part a extends to a one-to-one
correspondence
$$
\hbox{affine varieties of $\C^n$} \longleftrightarrow\, \hbox{radical
ideals of $\C[x_1,\ldots,x_n]$}.
$$
\bigskip
\smallskip

\noindent \ul{C\sc{oordinate} R\sc{ings}}. We next consider polynomial
functions on an affine variety $V$.  Note that two polynomials $f,g
\in \C[x_1,\ldots,x_n]$ give the same function on $V$ if and only if
their difference lies in ${\bf I}(V)$.  Thus the ring of such
functions is naturally isomorphic to the quotient ring
$$
\C[V] = \C[x_1,\ldots,x_n]/{\bf I}(V).
$$
This ring is called the {\it coordinate ring\/} of $V$.  There is a
close relation between $V$ and $\C[V]$.  The following two exercises
explore aspects of this relation.
\bigskip

\ex{1.4} Two affine varieties $V_1 \subset \C^n$ and $V_2 \subset
\C^m$ are {\it isomorphic\/} if there are polynomial maps $F : \C^n
\to \C^m$ and $G: \C^m \to \C^n$ such that $F(V_1) = V_2$, $G(V_2) =
V_1$, and the compositions $F\circ G$ and $G\circ F$ are the identity
when restricted to $V_2$ and $V_1$ respectively.  Prove that two
affine varieties are isomorphic if and only if their coordinate rings
are isomorphic $\C$-algebras.
\bigskip

\ex{1.5} Let $V \subset \C^n$ be an affine variety.
\item{a.} Given $a = (a_1,\ldots,a_n) \in \C^n$, show that $a \in V$
if and only if ${\bf I}(V) \subset \langle
x_1-a_1,\dots,x_n-a_n\rangle$.
\item{b.} Conclude that there is a one-to-one correspondence between
points of affine variety $V$ and maximal ideals of its coordinate ring
$\C[V]$. 
\bigskip

We can characterize coordinate rings of affine varieties as follows.

\proclaim Proposition 1.1.  A $\C$-algebra $R$ is isomorphic to the
coordinate ring of an affine variety if and only if $R$ is a finitely
generated $\C$-algebra with no nonzero nilpotents (i.e., if $f \in R$
satisfies $f^m = 0$, then $f = 0$).

\prf
If $R = \C[V]$ for $V \subset \C^n$, then we need only show that $R$
has no nonzero nilpotents.  This is easy, for if $f \in
\C[x_1,\ldots,x_n]$ and $f^m$ vanishes on $V$, then so does $f$.  Thus
${\bf I}(V)$ is radical, which means that $\C[V] =
\C[x_1,\ldots,x_n]/{\bf I}(V)$ has no nonzero nilpotents.

Conversely, $R$ finitely generated as a $\C$-algebra implies that
there is a surjective homomorphism $\varphi : \C[x_1,\ldots,x_n]
\to R$.  Let $I = \ker \varphi$, and note that $I = \sqrt{I}$ since $R$
has no nonzero nilpotents.  Then let $V = {\bf V}(I) \subset \C^n$.
The coordinate ring of $V$ is $\C[x_1,\ldots,x_n]/{\bf I}(V)$.  Using
the Nullstellensatz, we see that ${\bf I}(V) = {\bf I}({\bf V}(I)) =
\sqrt{I} = I$.  Thus $\C[V]$ is isomorphic to $R$.
\qed\medskip

To emphasize the close relation between $V$ and $\C[V]$, we will
sometimes write
$$
V = {\rm Spec}(\C[V]). \leqno(1.1)
$$
Furthermore, this can be made canonical by identifying $V$ with the
set of maximal ideals of $\C[V]$ via Exercise 1.5.  This is part of a
general contruction in algebraic geometry which takes any commutative
ring $R$ and defines the {\it affine scheme\/} {\rm Spec}(R).  The
general definition of Spec uses all prime ideals of $R$ and not just
the maximal ideals as we have done.\footnote*{Thus (1.1) should be
written $V = {\rm Specm}(\C[V])$, the maximal spectrum of $\C[V]$.}
Readers wishing to learn more about schemes should consult [4] and
[5].
\bigskip
\smallskip

\noindent \ul{S\sc{ubvarieties and the} Z\sc{ariski}
T\sc{opology}}. Given an affine variety $V \subset \C^n$, a subset $W
\subset V$ is a {\it subvariety\/} if $W$ is also an affine variety.
This easily implies that ${\bf I}(V) \subset {\bf I}(W)$.  In terms of
the coordinate ring $R = \C[V]$, we conclude that there is a
one-to-one correspondence
$$
\hbox{subvarieties of ${\rm Spec}(R)$} \longleftrightarrow\,
\hbox{radical ideals of $R$}.
$$

An affine variety has two interesting topologies.  First, we have the
induced topology from the usual topology on $\C^n$.  This is sometimes
called the {\it classical topology\/}.  The other topology is defined
as follows.  Given a subvariety $W \subset V$, the complement $V-W$ is
called a {\it Zariski open subset\/} of $V$.  One easily sees that the
Zariski open subsets of $V$ form a topology on $V$, which is called
the {\it Zariski topology\/}.  Since every subvariety of $V$ is closed
in the classical topology (polynomials are continuous), it follows
that every Zariski open subset is also open in the classical topology.
\bigskip

\ex{1.6} Zariski open subsets tend to be large.  Here are some
examples. 
\item{a.} Show that the Zariski topology on $\C$ is the {\it
cofinite\/} topology.  This is the topology whose open sets are
$\emptyset$ and complements of finite sets.
\item{b.} Show that the Zariski topology on $\C^n$ is $T_1$ but not
$T_2$.  
\bigskip

Given a subset $S \subset V$, its closure $\overline{S}$ in the
Zariski topology is the smallest subvariety of $V$ containing $S$.  We
call $\overline{S}$ the {\it Zariski closure\/} of $S$.  It is easy to
give examples where this differs from the closure in the classical
topology.

Finally, we remark that some Zariski open subsets of an affine variety
$V$ are themselves affine varieties.  Given $f \in \C[V] - \{0\}$, let
$V_f = \{a \in V \mid f(a) \ne 0\} \subset V$.  

\proclaim Lemma 1.2. $V_f$ is Zariski open in $V$ and has a natural
structure as an affine variety.  

\prf Suppose $V \subset \C^n$ and ${\bf I}(V) = \langle
f_1,\dots,f_s\rangle$.  Also pick $g \in \C[x_1,\ldots,x_n]$ so that
$f = g + {\bf I}(V)$.  Then $V_f = V - {\bf V}(f_1,\ldots,f_s,g)$, so
that $V_f$ is Zariski open in $V$.

Consider a new variable $y$ and let $W = {\bf V}(f_1,\ldots,f_s,1-gy)
\subset \C^n\times\C$.  Then $(a,b) \in \C^n\times\C$ lies in $W$ if
and only if $a \in V_f$ (and then $b = 1/g(a)$).  In other words, the
projection map $\C^n\times\C \to \C^n$ maps $W$ bijectively to $V_f$.
Thus we can identify $V_f$ with the affine variety $W \subset
\C^n\times\C$.\qed
\bigskip
\smallskip

\noindent \ul{I\sc{rreducible} V\sc{arieties and} R\sc{ational}
F\sc{unctions}}.  An affine variety $V$ is {\it irreducible\/} if it
cannot be written as union of subvarieties $V = V_1 \cup V_2$ where
$V_i \ne V$.  We can think of irreducbility in algebraic terms as
follows.
\bigskip

\ex{1.7} Let $V \subset \C^n$ be an affine variety.  Prove that $V$ is
irreducible $\Leftrightarrow$ ${\bf I}(V) \subset \C[x_1,\dots,x_n]$
is a prime ideal $\Leftrightarrow$ the coordinate ring $\C[V]$ is an
integral domain.
\bigskip

Here is an example we will refer to later.
\bigskip

\ex{1.8} Let $V = {\bf V}(xy-zw) \subset \C^4$.
\item{a.} Show that $xy-zw$ is irreducible in $\C[x,y,z,w]$.
\item{b.} Conclude that ${\bf I}(V) = \langle xy-zw\rangle$ and that
$V$ is irreducible.  Thus the coordinate ring of $V$ is $\C[V] =
\C[x,y,z,w]/\langle xy-zw\rangle$.
\item{c.} Prove that $\C[V] \simeq \C[ab,cd,ac,bd] \subset
\C[a,b,c,d]$.  Hint: Prove that $V$ can be parametrized surjectively
by $(a,b,c,d) \mapsto (ab,cd,ac,bd)$.
\bigskip

When $V$ is irreducible, the integral domain $\C[V]$ has a field of
fractions denoted $\C(V)$.  This is the {\it field of rational
functions\/} on $V$.  For example, when $V = \C^n$, $\C[V]$ is the
polynomial ring $\C[x_1,\ldots,x_n]$ and $\C(V)$ is the field of
rational functions $\C(x_1,\ldots,x_n)$.  In general, given $f/g \in
\C(V)$, the equation $g = 0$ defines a proper subvariety $W \subset V$
and $f/g : V - W \to \C$ is a well-defined function.  This is written
$f/g : V -\!\to \C$ and is called a {\it rational function\/} on $V$.
\bigskip

\ex{1.9} If $V$ is irreducible and $f \in \C[V]$ is nonzero, then the
{\it localization\/} of $\C[V]$ at $f$ is
$$
\C[V]_f = \{g/f^\ell \in \C(V) \mid g \in \C[V],\ \ell \ge 0\}.
$$
Prove that ${\rm Spec}(\C[V]_f)$ is the affine variety $V_f$ from
Lemma~1.2.
\bigskip

An important result is that every affine variety $V$ can be written as
a union
$$
V = V_1 \cup \cdots \cup V_r
$$
where each $V_i$ is irreducible and $V_i \not\subset \bigcup_{j\ne i}
V_j$.  We call $V_1,\ldots,V_r$ the {\it irreducible components\/} of
$V$.  The existence and uniqueness of this decomposition is proved in
Chapter 4 of [3].

Finally, some references use different terminology.  For example, in
Hartshorne's book [5], ${\bf V}(I) \subset \C^n$ is called an
``algebraic set'' and the term ``affine variety'' is reserved for the
case when ${\bf V}(I)$ is irreducible.  We will not use this
terminology, though we should point out that our main objects of
interest are toric varieties, which are by definition irreducible.
\bigskip
\smallskip

\noindent \ul{N\sc{ormal} A\sc{ffine} V\sc{arieties}}. Let $R$ be an
integral domain with field of fractions $K$.  Then $R$ is {\it
integrally closed\/} if every element of $K$ which is integral over
$R$ (meaning that it is a root of a monic polynomial in $R[x]$)
actually lies in $R$.  Here are two examples:
\item{$\bullet$} One can easily show that any UFD is integrally
closed.
\item{$\bullet$} The set ${\cal O}_K$ of all algebraic integers in a
number field $K$ is integrally closed.
\vskip0pt
\noindent Exercise 1.10 below will give an example of an integral
domain which is not integrally closed.

Let $V$ be an irreducible affine variety, so that $\C[V]$ is an
integral domain.  Then $V$ is {\it normal\/} if $\C[V]$ is integrally
closed.  For example, $\C^n$ is normal since its coordinate ring
$\C[x_1,\dots,x_n]$ is a UFD and hence integrally closed.  Here is an
example of a non-normal affine variety.
\bigskip

\ex{1.10} Let $C = {\bf V}(x^3-y^2) \subset \C^2$.  This is a plane
curve with a cusp at the origin.
\item{a.} Show that $C$ is irreducible and that $\C[C] =
\C[x,y]/\langle x^3-y^2\rangle$.
\item{b.} Let $X$ and $Y$ be the cosets of $x$ and $y$ in $\C[C]$
respectively.  This gives $Y/X \in \C(C)$.  Show that $Y/X \notin
\C[C]$ and that $(Y/X)^2 = X$.
\item{c.} Explain why part b implies that $\C[C]$ is not integrally
closed. 
\bigskip

Another example is the irreducible variety $V = {\bf V}(xy-zw) \subset
\C^4$ studied in Exercise~1.8.  It is not obvious, but $V$ is normal.
This can be proved using the description
$$
\C[V] \simeq \C[ab,cd,ac,bd] \subset \C[a,b,c,d]
$$
given in part c of Exercise 1.8.  The ring $\C[ab,cd,ac,bd]$ is a {\it
semigroup algebra\/}.  Then normality follows from a property called
{\it saturation\/}.

For us, normality is crucial because toric varieties are all normal.
(One can define non-normal toric varieties, but the nicest results
hold only in the normal case.)

Finally, any irreducible affine variety $V$ has a {\it
normalization\/}.  To define this, first consider
$$
\C[V]' = \{\alpha \in \C(V): \alpha\ \hbox{is
integral over}\ \C[V]\}.
$$
We call $\C[V]'$ the {\it integral closure} of $\C[V]$.  It is easy to
see that $\C[V]'$ is integrally closed.  With more work, one can also
show that $\C[V]'$ is a finitely generated $\C$-algebra (see Theorem~9
on pages~267--268 of [11]).  This gives the normal affine variety
$$
V' = {\rm Spec}(\C[V]')
$$
which is the {\it normalization\/} of $V$.  Note that the natural
inclusion $\C[V] \subset \C[V]' = \C[V']$ corresponds to a map $V' \to
V$.  This is called the {\it normalization map\/}.
\bigskip

\ex{1.11} Let $C = {\bf V}(x^3-y^2) \subset \C^2$ be the curve
considered in Exercise~1.10.
\item{a.} Let $X$ and $Y$ have the same meaning as in Exercise~1.10.
Show that $\C[Y/X] \subset \C(C)$ is the integral closure of $\C[C]$.
\item{b.} Show that the normalization map is the map $\C \to C$
defined by $t \mapsto (t^2,t^3)$.

\medskip
\bigskip


\noindent {\bigbf \S2. 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
$$
(a_0,\ldots,a_n) \sim (b_0,\ldots,b_n) \iff \hbox{there is } \lambda
\in \C^*\ \hbox{with}\ (a_0,\ldots,a_n) =
\lambda(b_0,\ldots,b_n).\leqno(2.1)  
$$
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{2.1} $\P^n$ 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 2.1 shows that $\P^n$ contains an isomorphic copy of
$(\C^*)^n$.  $\P^n$ is a classic example of a toric variety.

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

\ex{2.2} 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 (2.1).  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 
$$
f(\lambda x_0,\ldots, \lambda x_n) = \lambda^d
f(x_0,\ldots,x_n). \leqno(2.2) 
$$
\smallskip

\ex{2.3} 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{2.4} 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 a 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{2.5} 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 2.5 is actually part of the projective version of the
Nullstellensatz, which goes as follows.  We refer the reader to [3,
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 (2.2) shows that we get a
well-defined function
$$
{f\over g} : \P^n -{\bf V}(g) \longrightarrow \C
$$
As in \S1, 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{2.6} The set of all rational functions on $\P^n$ is
$$
\C(\P^n) = \Big\{{f\over g} \mid f,g \in \C[x_0,\ldots,x_n]\
\hbox{homogeneous of equal degree,}\ g \ne 0\Big\}. 
$$
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{2.7} 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 map $(a_0,\ldots,a_n) \mapsto
(f_0(a_0,\ldots,a_n),\ldots,f_m(a_0,\ldots,a_n))$ induces a
well-defined function $F : V \longrightarrow \P^m$.
\bigskip

An important fact is that in the situation of Exercise 2.7, the image
$F(V) \subset \P^m$ is a projective subvariety.  When $V = \P^n$, this
is proved in [3, Chapter 8], and the proof extends easily to cover the
general case.
\bigskip

\ex{2.8} When $V \subset \C^n$ is an affine variety 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{2.9} 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~2.9 shows that projective varieties are unions of affine
varieties.
\bigskip

\ex{2.10} 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~2.9, $V\cap U_i$ 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 2.10 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{2.11} 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~2.6.
\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{2.12} 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
$$
f(\lambda^{q_0} x_0,\ldots, \lambda^{q_n} x_n) = \lambda^d
f(x_0,\ldots,x_n). \leqno(2.3) 
$$
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{2.13} 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{2.14} 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 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{2.15} 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 $(a,b,c) \mapsto (a^2/c,ab/c,b^2/c)$.
\bigskip

One shows that $\P(1,1,2)$ is the abstract variety obtained by
``gluing'' two copies of $\C^2$ together with the affine variety $U_2$
from part c of Exercise~2.15.  But we must first understand what
``gluing'' means.
\bigskip
\medskip

\noindent {\bigbf \S3. 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 Hausdorff 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
$$
\cO_V^\infty(U) = \{f : U \to \R \mid f\ \hbox{is a}\ C^\infty\
\hbox{function}\}.\leqno(3.1)
$$

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 section but will switch to the
second for \S9.  When thinking in terms of sheaf cohomology,
one usually uses 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, (3.1) gives a ringed space over $\R$ denoted
$(V,\cO_V^\infty)$.
\bigskip

\ex{3.1} 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{3.2} 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{3.3} 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~3.3
is called the {\it structure sheaf\/} of $M$.
\bigskip

\ex{3.4} 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 \S1 generalize to an arbitrary affine variety $V
= {\rm Spec}(R)$.
\bigskip

\ex{3.5} 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 \S1, 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
\S1.
\bigskip

\ex{3.6} 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 [4] or [5] for the details of the proof.  These
references also show how to define $\cO_V$ when $V$ is not
irreducible.
\bigskip

\ex{3.7} 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 3.1. 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 3.1, 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 3.2. 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{3.8} 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{3.9} 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{3.10} 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~2.6 of \S2.
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 $\phi{\big|}_{U\setminus U\cap
{\bf V}(g)} = (f/g){\big|}_{U\setminus U\cap {\bf V}(g)}$.  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~2.11, 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~3.10, 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 [5, Appendix B] and [9] 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{3.11} 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{3.12} 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~3.12.

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 [4] and [5].  

\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 [4] and [5].    

\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~3.2 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{3.13} 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{3.14} 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{3.15} 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

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 [4] and [5].

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

\noindent {\bigbf \S4. 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{4.1} Consider the variety $X$ constructed by identifying two
copies of $\C$ along $\C-\{0\}$ (in the language of \S3, 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{4.2} 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{4.3} Prove that a subvariety of a separated variety is separated. 
\bigskip

Combining Exercises 4.2 and 4.3, we see that affine varieties are
always separated.  We will omit the proof that $\P^n$ is separated.
By Exercise~4.3, it follows that every projective variety is
separated.

In [9], Serre characterized separatedness in terms of the classical
topology as follows.

\proclaim Theorem 4.1.  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~4.1 is not Hausdorff (the two copies of the origin in $X$ do
not have disjoint neighborhoods).  This also gives a quick proof of
Exercise~4.2 since $\C^n$ is Hausdorff in the classical topology.

Here are some properties of separated varieties.

\proclaim Proposition 4.2.  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~3.15, 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\/}.  

\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 4.3. 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 4.4 below.\qed
\bigskip

\ex{4.4} 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.  

\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{4.5} Prove that a subvariety of a complete variety is complete.
\bigskip

\ex{4.6} Prove that the cartesian product of two complete varieties is
complete. 
\bigskip

\ex{4.7} 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 4.4. $\P^n$ is complete.

For a proof, note that Theorem 6 of [3, Chapter~8, \S5] implies that
the projection map $\P^n\times\C^m \to \C^m$ is closed.  By
Exercise~4.7, it follows that $\P^n$ is complete, and then any
projective variety is complete by Exercise~4.5.

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
$$
f_1 = \cdots = f_s = 0\leqno(4.1)
$$
have a nontrivial solution in the $x_i$?

To answer this question, observe that (4.1) defines a subvariety $W
\subset \P^n\times\C^m$, and the values of the $y_j$ for which (4.1)
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 (4.1) has a
nontrivial solution for the parameter values $y_j = b_j$ if and only
if
$$
g_1(b_1,\ldots,b_m) = \cdots = g_l(b_1,\ldots,b_m) = 0.\leqno(4.2)
$$
Chapter 8 of [3] gives an algorithm for finding the polynomials $g_i$.
We say that (4.2) is obtained from (4.1) by {\it eliminating\/} the
variables $x_0,\ldots,x_n$.  This is {\it projective elimination
theory\/}.

Serre's paper [9] characterizes completeness in terms of the classical
topology as follows.

\proclaim Theorem 4.5. 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~2.2).  In algebraic
geometry, completeness is a very useful property.  Here is a result
which indicates some reasons why.

\proclaim Theorem 4.6. 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 [4] and [5] for the
full definition.  Turning to Serre's classic paper [9] yet again, we
can characterize properness in terms of the classical topology as
follows.

\proclaim Theorem 4.7. 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{4.8} 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 \S1, we will need the following exercise.
\bigskip

\ex{4.9} 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 4.8.  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 \S1 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 4.9 implies that
$\cO_{V,p}$ is also integrally closed.  Conversely, if all of the
$\cO_{V,p}$ are integrally closed, then one easily shows that
$$
\bigcap_{p\in V} \cO_{V,p}
$$
is also integrally closed.  However, this intersection is precisely
$\cO_V(V)$, which equals $R$ by part a of Theorem 3.1 of \S3.
It follows that $R$ is integrally closed.\qed
\medskip

Proposition 4.8 has a following immediate corollary.

\proclaim Corollary 4.9. 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
\medskip


\noindent {\bigbf \S5. Smooth and Quasismooth Varieties}
\bigskip
\smallskip

\noindent \ul{T\sc{he} D\sc{imension of a} V\sc{ariety}}.  We
were using a very naive notion of {\it dimension\/} when we asserted
that $\C^n$ and $(\C^*)^n$ have dimension $n$.  For an arbitrary
variety $X$, there are several ways to define $\dim X$ rigorously.  In
the irreducible case, we do this as follows.

\proclaim Definition 5.1. The {\bf dimension\/} of an irreducible
variety $X$ is:
\item{$\bullet$} The transcendence degree of $\C(X)$ $($this is the maximal
number of algebraically independent elements of $\C(X)$$)$.
\item{$\bullet$} The maximum number $n$ such that one can find distinct
irreducible subvarieties
$$
\emptyset \ne V_0 \subset V_1 \subset \cdots \subset V_n = X.
$$

It is not at all obvious that these definitions coincide, but they
do---see [5].  In the affine or projective case, one can also define
dimension using the degree of an appropriate Hilbert polynomial.  This
approach is used in [3].  

Some important results concerning the dimension of a variety are:
\item{$\bullet$} $\C^n$, $(\C^*)^n$ and $\P^n$ have dimension $n$.
\item{$\bullet$} The dimension of a variety is the maximum of the
dimensions of its irreducible components.
\item{$\bullet$} If $W$ is a subvariety of $V$, then $\dim W \le \dim
V$.  Furthermore, if $V$ is irreducible and $W$ is a proper
subvariety, then $\dim W < \dim V$.
\item{$\bullet$} $\dim X\times Y = \dim X + \dim Y$.
\item{$\bullet$} If $V$ is an irreducible affine variety and $f \in
\C[V]$ is not invertible, then every irreducible component of ${\bf
V}(f) \subset V$ has codimension 1.
\item{$\bullet$} Let $V \subset \P^n$ be irreducible of positive
dimension and pick $f \in \C[x_0,\ldots,x_n]$.  If $f$ doesn't vanish
on $V$, then then every irreducible component of $V\cap {\bf V}(f)
\subset V$ has codimension 1.
\bigskip
\smallskip

\noindent \ul{T\sc{he} D\sc{imension of a} V\sc{ariety at a}
P\sc{oint}}.  The dimension of a variety $X$ at a point $p \in X$,
denoted $\dim_p X$, is defined as either:
\item{$\bullet$} The maximum of the dimensions of the irreducible
components of $X$ which contain $p$. 
\item{$\bullet$} The Krull dimension of the local ring $\cO_{X,p}$
(this is one less than the maximum length of a chain of prime ideals
in $\cO_{X,p}$).
\vskip0pt
\noindent Some of the properties of the dimension at a point include:
\item{$\bullet$} $\dim X = \max_{p\in X} \dim_p X$.
\item{$\bullet$} If $p \in Y \subset X$, then $\dim_p Y \le \dim_p
X$. 
\item{$\bullet$} If $p \in X$ and $q \in Y$, then $\dim_{(p,q)}
X\times Y = \dim_p X + \dim_p Y$. 

\smallskip
\bigskip
\noindent \ul{T\sc{he} Z\sc{ariski} T\sc{angent} S\sc{pace}}.  In
multivariable calculus, one defines the tangent space at a point of a
surface in $\R^3$, and this generalizes to the tangent space at a
point of a differentiable manifold.  In algebraic geometry, the {\it
Zariski tangent space\/} plays a similar role.

\proclaim Definition 5.2. Let $p$ be a point of a variety $X$ and let
$\frakm_{X,p}$ be the maximal ideal of the local ring $\cO_{X,p}$.
Then the {\bf Zariski tangent space} is defined to be
$$
T_p(X) = {\rm Hom}_\C(\frakm_{X,p}/\frakm_{X,p}^2,\C).
$$
\medskip

\ex{5.2} Use $\cO_{X,p}/\frakm_{X,p} \simeq \C$ to prove that
$\frakm_{X,p}/\frakm_{X,p}^2$ has a natural structure as a vector
space over $\C$.  This shows that Definition~5.2 makes sense.  
\bigskip

\ex{5.3} Let $V \subset \C^n$ be an affine variety and $p  =
(a_1,\ldots,a_n) \in V$.  
\item{a.} Show that $\frakm_{\C^n\!,p} = \langle
x_1-a_1,\ldots,x_n-a_n\rangle \subset \cO_{\C^n\!,p}$.
\item{b.} Show that $\frakm_{\C^n\!,p}/\frakm_{\C^n\!,p}^2$ has
dimension $n$, and conclude that $\dim_\C T_p(\C^n) = n$.
\item{c.} Use the surjection $\cO_{\C^n\!,p} \to \cO_{V,p}$ to
construct natural inclusion $T_p(V) \subset T_p(\C^n)$.
\item{d.} Conclude that $\dim_\C T_p(V) \le n$.
\bigskip

In the affine case, page 32 of [5] shows how to compute the Zariski
tangent spaces as follows.

\proclaim Lemma 5.3.  Let $V  \subset \C^n$ be a affine variety and
let $p \in V$.  Also assume that ${\bf I}(V) = \langle
f_1,\ldots,f_s\rangle$.  For each $i$, let
$$
d_p(f_i) = {\partial f_i\over \partial x_1}(p)\, x_1 + \cdots +
{\partial f_i\over \partial x_n}(p)\, x_n.
$$
The $T_p(V)$ is isomorphic to the subspace of $\C^n$ defined by
$d_p(f_1) = \cdots = d_p(f_s) = 0$.

\ex{5.4} Let $V = {\bf V}(x^3-y^2) \subset \C^2$.  For each $p \in V$,
show that $\dim_\C T_p(V) = 1$ unless $p$ is the origin, in which case
the dimension is 2.
\bigskip

\ex{5.5} If $p \in X$ and $q \in Y$, prove that 
$T_{(p,q)}(X\times Y) \simeq T_p(X) \oplus T_q(Y)$.  Hint:~Reduce to
the affine case and use Lemma 2.3.  See also Exercise~3.15 of \S3.
\bigskip

In general, we always have $\dim_\C T_p(X) \ge \dim_p X$. See Exercise
5.10 of [5, Chapter~I].

\smallskip
\bigskip
\noindent \ul{S\sc{mooth} V\sc{arieties}}.  As with dimension, there
are many ways to define smoothness.  

\proclaim Definition 5.4. A variety $X$ is {\bf smooth} or {\bf
nonsingular} at $p \in X$ if $\dim_\C T_p(X) = \dim_p(X)$.  We say
that $p$ is a {\bf singular} point of $X$ if it is not a smooth point.

Since $T_p(X) = {\rm Hom}_\C(\frakm_{X,p}/\frakm_{X,p}^2,\C)$, we see
that $X$ is smooth at $p$ when $\dim_p(X)$ equals the dimension of
$\frakm_{X,p}/\frakm_{X,p}^2$ as a vector space over
$\cO_{X,p}/\frakm_{X,p}$.  In terms of commutative algebra, this means
that $p \in X$ is smooth if and only if $\cO_{X,p}$ is a regular local
ring.  

By Exericse~5.3, every point of $\C^n$ is smooth (such a variety is
called {\it smooth\/}).  For a point of a subvariety of $\C^n$, we can
test for smoothness as follows.
\bigskip

\ex{5.6} Let $V \subset \C^n$ be an affine variety and let ${\bf I}(V)
= \langle f_1,\ldots, f_s\rangle$.  Also let $p \in V$ and set $d =
\dim_p V$.  Then prove that $V$ is smooth at $p$ if and only if the
Jacobian matrix
$$
J_p(f_1,\ldots,f_s) = \Big({\partial f_i\over \partial
x_j}(p)\Big)_{1\le i \le s,1\le j\le n}
$$
has rank $n-d$.  Hint:~Use Lemma 5.3.
\bigskip

\ex{5.7} Let $V = {\bf V}(xy-zw)$.  
Prove that the origin is the only singular point of $V$.
\bigskip

\ex{5.8} Let $p \in X$ and $q \in Y$.  Prove that $X\times Y$ is
smooth at $(p,q)$ if and only if $X$ and $Y$ are smooth at $p$ and $q$
respectively.  Hint:~Use Exercise~5.5 and the properties of $\dim_p
X$. 
\bigskip

\ex{5.9} Given a variety $X$, the set $X_{\rm sing} = \{p \in X \mid p\
\hbox{is singular}\}$ is the {\it singular locus\/} of $X$.  Use
Exercise~5.5 to prove that $X_{\rm sing}$ is a subvariety of $X$.
(With more work, one can show that $X_{\rm sing}$ is a proper
subvariety of $X$.  See Theorem~5.3 of [5].)

\smallskip
\bigskip
\noindent \ul{L\sc{ocal} A\sc{nalytic} E\sc{quivalence}}. When we want
to say that two varieties are {\it locally\/} the same, we have to be
careful to specify what we mean by ``local''.  

Suppose that we have $p \in X$ and $q \in Y$, where $X$ and $Y$ are
varieties.  Then, in the Zariski topology, $X$ and $Y$ being ``locally
equivalent'' at $p$ and $q$ respectively means that there are Zariski
open sets $p \in U \subset X$ and $q \in V \subset Y$ such that $U
\simeq V$ as varieties.  Since Zariski open subsets are huge, this
notion of ``local equivalence'' is not very useful.
\bigskip

\ex{5.10} Show that $p \in X$ and $q \in Y$ are ``locally equivalent''
in the above sense if and only if the local rings $\cO_{X,p}$ and
$\cO_{Y,q}$ are isomorphic as $\C$-algebras.
\bigskip 

In \S3, we discussed the sheaf $\cO_X^{\rm an}$ of analytic
functions on $X$, which is a sheaf in the classical topology.  This
allows one to define an analytic (or holomorphic) map between
classical open sets in varieties.  Then $X$ and $Y$ are {\it
analytically equivalent\/} at $p$ and $q$ if there are classical
open sets $p \in U \subset X$ and $q \in V \subset Y$ such that $U
\simeq V$ as analytic varieties.  Here are two nice facts about local
analytic equivalence:
\item{$\bullet$} As in Exercise~5.7, $X$ and $Y$ are analytically
equivalent at $p$ and $q$ if and only if there is a $\C$-algebra
isomorphism between the local rings $\cO_{X,p}^{\rm an}$ and
$\cO_{Y,q}^{\rm an}$.
\item{$\bullet$} $p \in X$ is smooth if and only if it is analytically
equivalent to $0 \in \C^n$.
\vskip0pt

\smallskip
\bigskip
\noindent \ul{F\sc{inite} Q\sc{uotients of} A\sc{ffine} S\sc{pace}}.
Let $G$ be a finite subgroup of ${\rm GL}(n,\C)$.  Then $G$
acts on $\C^n$, and the quotient $\C^n/G$ is the set of $G$-orbits.
By Chapter~7 of [3], we can turn this set into an affine variety as
follows.

\proclaim Proposition 5.5. Given a finite subgroup $G \subset {\rm
GL}(n,\C)$, let $\C[x_1,\ldots,x_n]^G \subset \C[x_1,\ldots,x_n]$ be
the subring of invariant polynomials.  There is a natural bijection
$\C^n/G \simeq {\rm Spec}(\C[x_1,\ldots,x_n]^G)$. 

Understanding the structure of $\C[x_1,\ldots,x_n]^G$ is one of the
goals of {\it invariant theory\/}.  In some cases, the quotient
$\C^n/G$ is still smooth.
\bigskip

\ex{5.11} Let $C_m \in {\rm GL}(n,\C)$ be the matrix with $e^{2\pi
i/m},1,\ldots,1$ on the main diagonal and 0's elsewhere, and let $G =
\{C_m^i \mid 0 \le i \le m-1\}$.  Use the map $\C^n \to \C^n$ given by
$(a_1,a_2,\ldots,a_n) \mapsto (a_1^m,a_2,\ldots,a_n)$ to prove that
$\C^n/G \simeq \C^n$.  Also, what is $\C[x_1,\ldots,x_n]^G$?
\bigskip

\ex{5.12} Let the symmetric group $S_n$ be embedded in ${\rm GL}(n,\C)$
as the set of permutation matrices.  Then $S_n$ acts on $\C^n$ by
permuting coordinates.  Prove that $\C^n/S_n \simeq \C^n$.
Hint:~Elementary symmetric polynomials.
\bigskip

A matrix in ${\rm GL}(n,\C)$ is a {\it complex reflection\/} if it is
conjugate to the matrix $C_m$ of Exercise~5.7, and $G \subset {\rm
GL}(n,\C)$ is a {\it complex rotation group\/} if it is generated by
complex rotations.  The Shephard-Todd-Chevalley theorem says that
$\C^n/G \simeq \C^n$ if and only if $G$ is a complex reflection group.
A proof can be found in [10, Section 2.4].
\bigskip

\ex{5.13} Show that the $n\times n$ permutation matrices form a
complex reflection group. 
\bigskip

\ex{5.14} Let $G \subset {\rm GL}(n,\C)$ be a finite subgroup and let
$H$ be subgroup of $G$ generated by the elements of $G$ which are
complex reflections.  Prove that $H$ is normal in $G$.
\bigskip

We next define a special type of finite matrix group.

\proclaim Definition 5.6. A finite subgroup $G \subset {\rm GL}(n,\C)$
is {\bf small\/} if it contains no complex reflections other than the
identity.

Small subgroups were introduced by Prill in order to obtain a
one-to-one correspondence between groups and quotients.  More
precisely, we have the following results, proved in [7]:
\item{$\bullet$} If $G \subset {\rm GL}(n,\C)$ is finite, then in a
classical neighborhood of the origin, $\C^n/G$ is analytically
equivalent to the quotient of $\C^n$ by a small subgroup.  (The rough
idea is that if $H$ is the subgroup of Exericise~5.14, then $\C^n/H
\simeq \C^n$ by the Shephard-Todd-Chevalley Theorem, and in a
classical neighborhood of the origin, the action of $G/H$ on $\C^n$ is
analytically equivalent to the linear action of a small subgroup.)
\item{$\bullet$} If $G_1$ and $G_2$ are small subgroups of ${\rm
GL}(n,\C)$ which give analytically equivalent singularities, then
$G_1$ and $G_2$ are conjugate in ${\rm GL}(n,\C)$.

\smallskip
\bigskip
\noindent \ul{Q\sc{uasimooth} V\sc{arieties}}.  We now define a
type of singularity which is close to being smooth.

\proclaim Definition 5.7. A point $p$ of a variety $X$ is a {\bf
finite quotient singularity} if there is a small subgroup $G \subset
{\rm GL}(n,\C)$ such that $p \in X$ is analytically equivalent to $0
\in \C^n/G$.  Then $X$ is {\bf quasismooth} or has {\bf finite
quotient singularities} or is {\bf $\Q$-smooth} if every point of $p$
is a finite quotient singularity.

Note that the definition of finite quotient singularity allows $G$ to
be the trivial subgroup of ${\rm GL}(n,\C)$.  It follows that any
smooth variety is quasismooth.  Here is an example to show that the
converse is not true.
\bigskip

\ex{5.15} Let $V = {\bf V}(xz-y^2) \subset \C^3$.  
\item{a.} Show that the origin is the unique singular point of $V$.
\item{b.} Let $G = \{\pm I\} \subset {\rm GL}(2,\C)$.  If we think of
$\C^2$ as ${\rm Spec}(\C[a,b])$, then show that $\C[a,b]^G =
\C[a^2,ab,b^2]$. 
\item{c.} Show that $\C[a^2,ab,b^2] \simeq \C[x,y,z]/\langle
xz-y^2\rangle$, and conclude that $\C^2/G \simeq V$.
\bigskip

For the surface $V \subset \C^3$ of this exercise, $0 \in V$ is not
smooth by part a and is a finite quotient singularity by part c.
Since all other points of $V$ are smooth, we see that $V$ is
quasismooth but not smooth.

We can generalize Exercise 5.15 as follows.

\proclaim Proposition 5.8.  Let $G \subset \C^n$ be a small subgroup.
Then $\C^n/G$ is quasismooth.

\prf The definition of quasismooth guarantees that $0 \in \C^n/G$ is a
finite quotient singularity.  But what about the other points of
$\C^n/G$?  Given $v \in \C^n$, let $G_v = \{g \in G \mid g\cdot v =
v\}$ be its isotropy subgroup.  We will show that $v \in \C^n/G$ is
analytically equivalent to $0 \in \C^n/G_v$.

First observe that $w \mapsto w + v$ is equivariant with respect to
the action of $G_v$, as is $w \mapsto w-v$.  This gives an
isomorphism of varieties $\C^n/G_v \to \C^n/G_v$ which takes 0 to $v$.
Thus $0 \in \C^n/G_v$ is analytically equivalent to $0 \in \C^n/G_v$.

Hence we need only show that $v \in \C^n/G_v$ is analytically
equivalent to $v \in \C^n/G$.  Let $\{g_i\}$ be left coset
representatives for $G/G_v$.  Then $\C^n/G$ is obtained from
$\C^n/G_v$ by identifying $w \in \C^n/G_v$ with $g_i\cdot w$ for all
$i$.  Since the points $g_i\cdot v$ are distinct in $\C^n/G_v$, we can
find a classical neighborhood $U$ of $v \in \C^n/G_v$ such that the
neighborhoods $g_i\cdot U$ are disjoint.  The $g_i$ act on $\C^n/G_v$
as isomorphisms of varieties, which implies that $v \in U \subset
\C^n/G_v$ is analytically equivalent to a neighborhood of $v \in
\C^n/G$.  This gives the desired analytic equivalence.\qed 
\bigskip

\ex{5.16} Prove that a cartesian product of quasismooth varieties is
quasismooth.
\bigskip
\medskip

\noindent {\bigbf \S6. The Local Ring of an Irreducible Hypersurface}
\bigskip

Let $X$ be an irreducible variety with function field $\C(X)$.  A
subvariety $Y \subset X$ is a {\it hypersurface\/} if every
irreducible component of $Y$ has codimension 1 in $X$. 
\bigskip
\smallskip

\noindent \ul{The Local Ring}.  Let $Y \subset X$ be an irreducible
hypersurface.  Then consider the set 
$$
\cO_{X,Y} = \{f \in \C(X) \mid f\ \hbox{is defined on a nonempty
Zariski open subset of}\ Y\}.
$$
To understand this, recall that $f \in \C(X)$ means that there is a
nonempty Zariski open $Y \subset X$ and $f : U \to \C$ is a morphism.
Then $f \in \cO_{X,Y}$ when we can find such a $U$ satisfying $U\cap Y
\ne 0$.
\bigskip

\ex{6.1} Prove that $\cO_{X,Y}$ is a local ring and that the maximal
ideal consists of those $f \in \cO_{X,Y}$ which vanish on $Y$.
\bigskip

\ex{6.2} Let $Y = {\bf V}(x) \subset \C^2$.  
\item{a.} Prove that
$$
\cO_{\C^2,Y} = \Big\{{P(x,y)\over Q(x,y)} \mid P(x,y),Q(x,y) \in
\C[x,y],\ Q(0,y) \ne 0\Big\}. 
$$
\item{b.} Given $f \in \C(x,y)$, prove that $f = x^m g$, where $m
\in \Z$ and $g \in \cO_{\C^2,Y}$ is a unit.  Hint: Write $f = P/Q$,
where $Q(0,y) \ne 0$.  Explain why $P = x^kP'$ and $Q = x^lQ'$, where
$P(0,y)$ and $Q(0,y)$ are nonzero.
\item{c.} Prove that every nonzero ideal of $\cO_{\C^2,Y}$ is of the
form $\langle x^m\rangle$ for some $m \ge 0$. 
\bigskip

Given $f \in \C(x,y)$, Exercise 6.2 tells us that $f = x^m g$ for $m
\in \Z$ and $g$ a unit in $\cO_{\C^2,Y}$.  We call $m$ the {\it order
of vanishing} of $f$ on $Y = {\bf V}(x) \subset \C^2$ and denote it by
${\rm ord}_Y(f)$.
\bigskip
\smallskip

\noindent \ul{Discrete Valuation Rings}.  The crucial observation is
that Exercise 6.2 generalizes to any normal variety.  Let $R$ be an
integral domain with field of fractions $K$, and set $K^* = K -
\{0\}$.  Then $R$ is a {\it discrete valuation
ring\/} if there is a surjective function
$$
{\rm ord}_R : K^* \to \Z
$$
such that every for $a,b \in  K^*$, we have: 
\item{$\bullet$} ${\rm ord}_R(ab) = {\rm ord}_R(a) + {\rm ord}_R(b)$.
\item{$\bullet$} ${\rm ord}_R(a+b) \ge \min({\rm ord}_R(a),{\rm
ord}_R(b))$ provided $a+b \ne 0$.
\item{$\bullet$} $R = \{a \in K^* \mid {\rm ord}_R(a) \ge 0\} \cup
\{0\}$.
\vskip0pt
\noindent We say that ${\rm ord}_R$ is a {\it valuation\/} on $K$ and
that $R$ is its {\it valuation ring\/}.
\bigskip

\ex{6.3} Let $R$ be a discrete valuation ring.  
\item{a,} Prove that $R$ is a local ring with $\frakm = \{a \in R \mid
{\rm ord}_R(a) > 0\}$ as maximal ideal.
\item{b.} Let $a \in R$ satisfy ${\rm ord}_R(a) = 1$ ($a$ exists
because ${\rm ord}_R$ is onto).  Prove that $\frakm = \langle
a\rangle$. 
\item{c.} Let $a$ be as in part b.  Prove that any nonzero ideal of
$R$ is of the form $\langle a^m\rangle$ for some $m \ge 0$.
\bigskip

\ex{6.4} Prove that a discrete valuation ring is an integrally closed
one-dimensional Noetherian local ring.  (A ring $R$ is {\it
Noetherian\/} if every ideal of $R$ is finitely generated, i.e., if
the Hilbert Basis Theorem holds for $R$.)
\bigskip

Here are two classic examples of discrete valuation rings.
\bigskip

\ex{6.5} Let $p$ be prime.  Prove that $\Z_{(p)} = \{a/b \mid a,b \in
\Z,\ \gcd(p,b) = 1\}$ is a discrete valuation ring.  This gives the
{\it $p$-adic valuation\/}, denoted ${\rm ord}_p$.
\bigskip

\ex{6.6} Let $\C\{\!\{z\}\!\}$ the ring of complex power series with
positive radius of convergence.  Prove that $\C\{\!\{z\}\!\}$ is a
discrete valuation ring and that the valuation gives the order of
vanishing of a nonzero element of $\C\{\!\{z\}\!\}$.
\bigskip

For us, the main result we need is as follows.

\proclaim Theorem 6.1. Let $Y$ be an irreducible hypersurface in  a
normal variety $X$.  Then $\cO_{X,Y}$ is a discrete valuation ring.

\prf
The argument requires substantial amounts of commutative algebra.  We
will omit the details and just sketch the ideas involved.  One begins
with the following observations:
\item{$\bullet$} $\cO_{X,Y}$ is integrally closed since the
localization of a integrally closed domain is integrally closed.
\item{$\bullet$} $\cO_{X,Y}$ has dimension 1 as a ring since $Y$
having codimension 1 in $X$.
\item{$\bullet$} $\cO_{X,Y}$ is Noetherian since the localization of a
Noetherian ring is Noetherian. 
\vskip0pt
\noindent Thus $\cO_{X,Y}$ is a integrally closed one-dimensional
Noetherian local ring.  A classic result states that any such ring is
a discrete valuation ring (and conversely, as you showed in
Exercise~6.4).  The commutative algebra used here can be found in [1],
especially Chapter 9.
\qed
\medskip

In the situation of Theorem 6.1, the corresponding valuation is
written
$$
{\rm ord}_Y : \C(X)^* \to \Z.
$$
Given $f \in \C(X)^*$, we say that $f$ {\it vanishes to order\/} $m$
along $Y$ if $m = {\rm ord}_Y(f) > 0$ and {\it has a pole of order\/}
$m$ on $Y$ if $m = -{\rm ord}_Y(f) > 0$.
\medskip
\bigskip

\noindent {\bigbf \S7. Weil Divisors on Normal Varieties}
\bigskip
\smallskip

\noindent \ul{Weil Divisors}. A {\it Weil divisor\/} on a normal
variety $X$ is a finite formal sum
$$
D = \sum_{i=1}^s a_i D_i
$$
where the $D_i$ are distinct irreducible hypersurfaces of $X$ and $a_i
\in \Z$.   The set of all Weil divisors is a group under addition and is
denoted
$$
{\rm WDiv}(X).
$$
We say that $D = \sum_{i=1}^s a_i D_i$ is {\it effective\/} if $a_i
\ge 0$ for all $i$, and we write this as 
$$
D \ge 0.
$$  
Note that any Weil divisor can be written uniquely as the difference
of two effective Weil divisors.
\bigskip
\smallskip

\noindent \ul{The Divisor of a Rational Function}.  Given $f \in
\C(X)$, we can define ${\rm ord}_Y(f)$ for every irreducible
hypersurface $Y \subset X$.  This gives a Weil divisor as follows.

\proclaim Proposition 7.1.  Let $X$ be normal and $f \in \C(X)$ be
nonzero.  Then there are at most finitely many hypersuraces $Y \subset
X$ such that ${\rm ord}_Y(f) \ne 0$.  Thus we can define the Weil
divisor
$$
{\rm div}(f) = {\textstyle \sum_Y} {\rm ord}_Y(f)\, Y.
$$

\prf Let $U \subset \C$ be a Zariski open where $f : U \to \C$ is a
nonzero morphism.  Then $U' = U - f^{-1}(0)$ is also Zariski open in
$X$.  If $Y \subset X$ is an irreducible hypersurface with $Y \cap U'
\ne \emptyset$, then ${\rm ord}_Y(f) = 0$ since $f$ is defined but
nonvanishing on $Y\cap U'$.  Thus ${\rm ord}_Y(f) \ne 0$ implies
$Y \subset X - U'$.  Since $X-U'$ is a proper subvariety of $X$ and
$Y$ has codimension 1, it follows that $Y$ must be an irreducible
component of $X-U'$.  Then we are done since $X-U'$ has at most
finitely many irreducible components. \qed
\medskip

We sometimes write ${\rm div}(f) = {\rm div}_0(f) - {\rm
div}_\infty(f)$, where
$$
\eqalign{{\rm div}_0(f) &= \sum_{{\rm ord}_Y(f) > 0} {\rm ord}_Y(f)\,
		Y\cr
	{\rm div}_\infty(f) &= \sum_{{\rm ord}_Y(f) < 0} -{\rm
		ord}_Y(f)\, Y.\cr}
$$
We call ${\rm div}_0(f)$ (resp.\ ${\rm div}_\infty(f)$) the {\it
divisor of zeros\/} of $f$ (resp.\ the {\it divisor of poles\/} of
$f$).  Note that these are effective divisors.
\bigskip

\ex{7.1} Explain why ${\rm div}(fg) = {\rm div}(f) + {\rm div}(g)$ and
${\rm div}(1/f) = -{\rm div}(f)$ for $f,g \in \C(X)^*$.
\bigskip

\ex{7.2} Let $f \in \C[t]$ be a polynomial of degree $n$, and write $f
= c (x-a_1)^{m_1}\cdots (x-a_r)^{m_r}$, where $a_1,\ldots,a_r \in \C$
are distinct.
\item{a.} When $X = \C$, show that ${\rm div}(f) = \sum_{i=1}^r m_i\,
\{a_i\}$. 
\item{b.} When $X = \P^1 = \C\cup\{\infty\}$, show that ${\rm div}(f)
= \sum_{i=1}^r m_i\,\{a_i\} - n\{\infty\}$. 
\bigskip

Finally, we need to know when the divisor of a rational function
vanishes. 

\proclaim Proposition 7.2. Let $X$ be a normal variety and let $f \in
\C(X)^*$.  Then ${\rm div}(f) \ge 0$ if and only if $f : X \to \C$
is a morphism, i.e., $f \in \cO_X(X)$.

\prf If $f : X \to \C$ is a morphism, then $f \in \cO_{X,Y}$ for every
$Y$, which in turn implies ${\rm ord}_Y(f) \ge 0$.  Hence ${\rm
div}(f) \ge 0$.  Going the other way, suppose that ${\rm div}(f) \ge
0$.  Then
$$
f \in {\textstyle \bigcap_Y} \cO_{X,Y},\leqno(7.1)
$$
where the intersection is over all irreducible hypersurfaces of $X$.
Hence $f$ is defined on a nonempty Zariski open subset of every
irreducible hypersurface.  It follows that $f$ is defined outside of
a subvariety of codimension at least 2.  Since $X$ is normal, a
standard result in commutative algebra implies that $f$ is defined
everywhere (see Exercise 7.3 below).\qed
\bigskip

\ex{7.3} Let $X = {\rm Spec}(R)$, where $R$ is integrally closed.  Let
$K$ be the fraction field of $R$ and suppose that $f \in K$ has ${\rm
div}(f) = 0$. 
\item{a.} Show that (7.1) implies that $f \in \bigcap_{\sfrakp}
R_{\sfrakp}$, where:
\itemitem{$\bullet$} The intersection is over all prime ideals $\frakp
\subset R$ such that ${\bf V}(\frakp)$ has codimension 1 in $X$ (these
are called prime ideal of {\it height\/} 1, written ${\rm ht}(\frakp)
= 1$).
\itemitem{$\bullet$} $R_{\sfrakp}$ is the localization of $R$ at the
multiplicative subset $R-\frakp$ (in Exercise~4.9 of \S4, this
was written $R_{R-\sfrakp}$).  
\vskip0pt
\item{b.} A theorem in commutative algebra states that $R =
\bigcap_{{\rm ht}(\sfrakp) = 1} R_{\sfrakp}$ whenever $R$ is
Noetherian and integrally closed.  A proof can be found in [6, \S12].
Explain how this completes the proof of Proposition 7.2.
\bigskip

While the proof of Proposition 7.2 uses a lot of commutative algebra,
there is also some nice intuition coming from several complex
variables.  Suppose that $U \subset \C^2$ is a classical neighborhood
of the origin and that $f$ is holomorphic on $U-\{(0,0)\}$.  Then {\it
Hartogs' Lemma\/} asserts that $f$ extends automatically to a
holomorphic function on $U$.  This applies more generally as
follows:~if $X$ is a normal analytic space and $f$ is homomorphic on
$X-Y$, where $Y$ has codimension at least 2, then $f$ extends to a
holomorphic function on $X$.

For a connected complete variety $X$, we learned in Theorem 4.6 of
\S4 that the only morphisms $X \to \C$ are constant.  This
gives the following corollary of Proposition 7.2.

\proclaim Corollary 7.3. Let $X$ be a complete normal variety and let
$f \in \C(X)^*$.  Then ${\rm div}(f) \ge 0$ if and only if $f$ is
constant.

Since ${\rm div}(1/f) = -{\rm div}(f)$ by Exercise 7.1, we see that
${\rm div}(f) = 0$ if and only if ${\rm div}(f) \ge 0$ and ${\rm
div}(1/f) \ge 0$.  Hence we get another corollary of Proposition~7.2.

\proclaim Corollary 7.4. Let $X$ be a normal variety and let $f \in
\C(X)^*$.  Then ${\rm div}(f) = 0$ if and only if $f : X \to \C^*$ is
a morphism, i.e., $f \in \cO_X(X)^*$ $($the group of invertible
elements of $\cO_X(X)$$)$.

\medskip

\noindent \ul{Linearly Equivalent Divisors and the Divisor Class
Group}.  As above, let $X$ be a normal variety.  We say that two Weil
divisors $D_1, D_2 \in {\rm WDiv}(X)$ are {\it linearly
equivalent\/}, written $D_1 \sim D_2$, if there is $f \in \C(X)^*$
such that ${\rm div}(f) = D_1 - D_2$.  Furthermore, we say that $D \in
{\rm WDiv}(X)$ is a {\it principal divisor\/} if $D \sim 0$, i.e., $D =
{\rm div}(f)$ for some $f \in \C(X)$.
\bigskip

\ex{7.4} Let $\sim$ be defined as above.
\item{a.} Use Exercise 2.1 to show that set of principal divisors is a
subgroup of ${\rm WDiv}(X)$.
\item{b.} Show that $\sim$ is an equivalence relation on ${\rm
Div}(X)$. 
\bigskip

The subgroup of principal divisors is denoted ${\rm Div}_0(X)$ (\S8
will explain this notation).  Parts a and b of Exercise~7.4 are
linked, of course, since $\sim$ is the equivalance relation coming
from the subgroup ${\rm Div}_0(X)$.  The quotient group
$$
{\rm Cl}(X) = {\rm WDiv}(X)/{\rm Div}_0(X)
$$
is the {\it divisor class group\/} of $X$.  It consists of equivalence
classes of linearly equivalent divisors.  Given $D \in {\rm
Div}(X)$, its divisor class in ${\rm Cl}(X)$ is denoted $[D]$.

Here is an important exact sequence involving the class group.
\bigskip

\ex{7.5} Let $X$ be a normal variety.  Use Corollary 7.4 to
prove that there is an exact sequence
$$
1 \to \cO_X(X)^* \to \C(X)^* \to {\rm WDiv}(X) \to {\rm Cl}(X) \to 0,
$$
where the map $\C(X)^* \to {\rm WDiv}(X)$ is $f \mapsto {\rm div}(f)$
and ${\rm WDiv}(X) \to {\rm Cl}(X)$ is $D \mapsto [D]$.
\bigskip

One pretty result we will need is the following.  A proof can be found
in Proposition 6.2 of [5, II.6].

\proclaim Theorem 7.5.  For a normal affine variety $X = {\rm
Spec}(R)$, the class group ${\rm Cl}(X)$ is trivial if and only if $R$
is a unique factorization domain.

The class group ${\rm Cl}(X)$ is sometimes denoted $A_{n-1}(X)$, where
$n = \dim X$.  More generally, one can define {\it Chow groups\/} for
$A_k(X)$ for any irreducible variety $X$.  

In another direction, let $\cO_K$ be the ring of algebraic integers in
a number field $K$.  Then the scheme $X = {\rm Spec}(R)$ is normal,
and ${\rm Cl}(X)$ can defined as above.  One can show that in this
case, ${\rm Cl}(X)$ is the {\it ideal class group\/} of $K$ as defined
in algebraic number theory.
\medskip
\bigskip

\noindent {\bigbf \S8. Cartier Divisors on Normal Varieties}
\bigskip

We will give a slightly non-standard treatment of Cartier divisors
which works nicely on normal varieties.
\smallskip
\bigskip

\noindent \ul{Our Definition}.  Let $D = \sum_{i=1}^s a_i D_i$ be a
Weil divisor on a normal variety $X$.  If $U \subset X$ is a nonempty
Zariski open subset, then the {\it restriction\/} of $D$ to $U$ is the
is Weil divisor
$$
D\big|_U = \sum_{U\cap D_i \ne \emptyset} a_i\, U\cap D_i.
$$
We now define a special class of Weil divisors.

\proclaim Definition 8.1.  Let $D$ be a Weil divisor on a normal
variety $X$.
\item{a.} $D$ is {\bf locally principal} if there is an open
cover $\{U_i\}_{i\in I}$ of $X$ such that $D\big|_{U_i}$ is principal
for every $i \in I$.
\item{b.} $D$ is {\bf Cartier} if it is locally principal.

A principal divisor is obviously locally principal.  Thus ${\rm
div}(f)$ is Cartier for all $f \in \C(X)^*$.
\bigskip

\ex{8.1} Let $D$ and $E$ be Cartier divisors.  Prove that $D+E$
and $-D$ are Cartier. 
\bigskip

\ex{8.2} Let $D \sim E$ be linearly equivalent Weil divisors.  Prove
that $D$ is Cartier if and only if $E$ is Cartier.
\bigskip

If $D$ is locally principal relative to the open cover $\{U_i\}_{i\in
I}$, then we can find $f_i \in \C(X)^*$ such that $D\big|_{U_i} = {\rm
div}(f_i)$ on $U_i$.  We say that $\{(U_i,f_i)\}_{i\in I}$ is {\it
local data\/} for $D$.
\bigskip

\ex{8.3} Let $\{(U_i,f_i)\}_{i\in I}$ be local data for a Cartier
divisor $D$.
\item{a.} Prove that $f_i/f_j \in \cO_X(U_i\cap U_j)^*$ for all $i,j
\in I$.  Hint:~Use Corollary~7.4.
\item{b.} Prove that $D$ is effective if and only if $f_i \in
\cO_X(U_i)$ for all $i \in I$.  Hint:~Use Proposition~7.2. 
\bigskip

For an example of a Weil divisor which is not Cartier, consider the
affine surface $X = {\bf V}(xy-z^2) \subset \C^3$.  The $x$-axis $Y =
{\bf V}(y,z)$ is contained in $V$, so that $Y$ is a Weil divisor on
$X$.  However, one can show that $Y$ is {\it not\/} a Cartier divisor
(see Example 6.11.3 in [5, II.6]).

There is one nice case where Weil and Cartier divisors coincide.

\proclaim Theorem 8.2. Let $X$ be a normal variety such that the local
ring $\cO_{X,p}$ is a unique factorization domain for every $p \in X$.
Then every Weil divisor on $X$ is Cartier.

This is proved in Proposition 6.11 of [5, II.6]).  We should
also mention that if $X$ is smooth, then $\cO_{X,p}$ is a unique
factorization domain for all $p$.  It follows that Weil and Cartier
divisors coincide on smooth varieties.  
\smallskip
\bigskip

\noindent \ul{The Standard Definition}.  Definition~8.1 differs from
what one usually finds in the literature.  The more common definition
starts with local data $\{(U_i,f_i)\}_{i\in I}$ satisfying part a of
Exercise~8.3 and defines two local data $\{(U_i,f_i)\}_{i\in I}$,
$\{(V_j,g_j)\}_{j\in J}$ to be equivalent if $f_i/g_j \in
\cO_X(U_i\cap V_j)^*$ for all $(i,j) \in I\times J$.  Then a {\it
Cartier divisor\/} is an equivalance class of local data.

There is also more sophisticated way to define Cartier divisors which
avoids equivalence classes.  We have the sheaf $\cO_X^*$ whose
sections over $U$ are the invertible elements in the ring $\cO_X(U)$,
and we can also regard $\C(X)^*$ as a constant sheaf on $X$.  Then one
can show that a Cartier divisor is a global section of the quotient
sheaf $\C(X)^*/\cO_X^*$.  See [5, page 141] for details.
\bigskip
\smallskip

\noindent \ul{The Picard Group}. We denote the set of all Cartier
divisors on a normal variety $X$ by
$$
{\rm Div}(X).
$$
This is a subgroup of ${\rm WDiv}(X)$ by Exercise 8.1.  Furthermore,
since every principal divisor is Cartier, we have ${\rm Div}_0(X)
\subset {\rm Div}(X)$.  Then we define the {\it Picard group\/} of $X$
to be the quotient
$$
{\rm Pic}(X) = {\rm Div}(X)/{\rm Div}_0(X).\leqno(8.1)
$$
We will give a more sophisticated definition of ${\rm Pic}(X)$ in \S10.
(Note that (8.1) explains why the group of principal divisors is
denoted ${\rm Div}_0(X)$ rather than ${\rm WDiv}_0(X)$.)  Since ${\rm
Div}(X)$ is a subgroup of ${\rm WDiv}(X)$, we get a canonical injection
$$
{\rm Pic}(X) \hookrightarrow {\rm Cl}(X).
$$
In analogy with Exercise 2.7, we have the following exact sequence.
\bigskip

\ex{8.4} Let $X$ be a normal variety.  Prove that there is an exact
sequence 
$$
1 \to \cO_X(X)^* \to \C(X)^* \to {\rm Div}(X) \to {\rm Pic}(X) \to 0,
$$
where the map $\C(X)^* \to {\rm Div}(X)$ sends $f$ to ${\rm div}(f)$
and the map ${\rm Div}(X) \to {\rm Pic}(X)$ is the natural
homomorphism.
\medskip
\bigskip

\noindent {\bigbf \S9. The Sheaf of a Weil Divisor}
\bigskip
\smallskip

\noindent \ul{Definition and Basic Properties}. Let $D$ be a Weil
divisor on a normal variety $X$.  We will show that $D$ determines a
sheaf $\cO_X(D)$ of $\cO_X$-modules on $X$.  As noted in \S3,
the sections of a sheaf ${\cal F}$ over $U \subset X$ can be written
$$
{\cal F}(U) = \Gamma(U,{\cal F}) = H^0(U,{\cal F}).
$$
For $\cO_X(D)$, we will find it convenient to use the middle notation.
Thus, given a Zariski open subset $U \subset X$, we define
$$
\Gamma(U,\cO_X(D)) = \{f \in \C(X)^* \mid ({\rm div}(f) + D)\big|_U
\ge 0\} \cup\{0\}.
$$

\proclaim Lemma 9.1. The above definition makes $\cO_X(D)$ into a
sheaf of $\cO_X$-modules on $X$.

\prf We first show that $\Gamma(U,\cO_X(D))$ is an additive subgroup
of $\C(X)$.  It suffices to prove this for $U = X$.  Let $D =
\sum_{i=1}^s a_i D_i$.  Then $f \in \Gamma(X,\cO_X(D))$ if and only if
${\rm ord}_{D_i}(f) \ge -a_i$ for all $i$.  If $g \in \C(X)^*$ also has
this property, then so does $f+g$ since
$$
{\rm ord}_{D_i}(f+g) \ge \min({\rm ord}_{D_i}(f),{\rm ord}_{D_i}(g))
\ge -a_i.
$$
Since ${\rm div}(-f) = {\rm div}(f)$, we see that $\Gamma(U,\cO_X(D))$
is a subgroup of $\C(X)$.

We next show that this is a module over $\Gamma(X,\cO_X) = \cO_X(X)$.
Given $f \in \Gamma(X,\cO_X(D))$ and $g \in \Gamma(X,\cO_X)$, we know
that ${\rm div}(f) + D \ge 0$ and ${\rm div}(g) \ge 0$. Then
$$
{\rm div}(gf) + D = {\rm div}(g) + {\rm div}(f) + D  \ge 0
$$
since a sum of effective divisors is effective.  This proves that $gf
\in \Gamma(X,\cO_X(D))$ and gives the desired module structure.

Finally, we omit the proof that $\cO_X(D)$ is a sheaf in the Zariski
topology.\qed
\bigskip

\ex{9.1} The trivial Weil divisor is denoted $0$.  Prove that
$\cO_X(0)$ coincides with the structure sheaf $\cO_X$.  Hint:~Use
Proposition~7.2.
\bigskip

We next show that linearly equivalent divisors give isomorphic
sheaves.

\proclaim Proposition 9.2. If $D \sim E$ are linearly equivalent
Weil divisors, then $\cO_X(D)$ and $\cO_X(E)$ are isomorphic as
sheaves of $\cO_X$-modules.

\prf
By assumption, we have $D = E + {\rm div}(g)$ for some $g \in
\C(X)^*$.  Then 
$$
\eqalign{f \in \Gamma(X,\cO_X(D)) &\iff {\rm div}(f) + D  \ge 0\cr
	&\iff {\rm div}(f) + E + {\rm div}(g) \ge 0\cr
	&\iff {\rm div}(fg) + E \ge 0\cr
	&\iff fg \in \Gamma(X,\cO_X(E)).\cr}
$$
Thus multiplication by $g$ induces an isomorphism $\Gamma(X,\cO_X(D))
\simeq \Gamma(X,\cO_X(E))$ which is clearly an isomorphism
of $\cO_X(X)$-modules.

The same argument works over any Zariski open set $U$, and the
isomorphisms are easily seen to be compatible with the restriction
maps.\qed
\bigskip
\smallskip

\noindent \ul{Weil Divisors on an Affine Variety}.  
Now suppose that $X = {\rm Spec}(R)$ is affine and let $K$ be the
field of fractions of $R$.  If $D$ is a Weil divisor on $X = {\rm
Spec}(R)$, then $\Gamma(X,\cO_X(D))$ is an $R$-submodule of $K$.  
We first prove that this $R$-module is finitely generated.

\proclaim Proposition 9.3. Let $D$ be a Weil divisor on the normal
affine variety $X = {\rm Spec}(R)$.  Then  $\Gamma(X,\cO_X(D))$ is a
finitely generated $R$-module.

\prf We will prove the existence of $h \in R-\{0\}$ such that
$h\Gamma(X,\cO_X(D)) \subset R$.  This will imply that
$h\Gamma(X,\cO_X(D))$ is an ideal of $R$ and hence has a finite basis
since $R$ is Noetherian.  It will follow immediately that
$\Gamma(X,\cO_X(D))$ is finitely generated as an $R$-module.

Write $D = \sum_{i=1}^s a_i D_i$.  Since $\bigcup_{i=1}^s D_i$ is a
proper subvariety of $X$, we can find $g \in R - \{0\}$ which vanishes
on each $D_i$.  Then ${\rm ord}_{D_i}(g) > 0$ for every $i$, so that
$m\,{\rm ord}_{D_i}(g) > a_i$ for all $i$, provided $m \in \Z$ is
sufficiently large.  Since ${\rm div}(g) \ge 0$, it follows that
$m\,{\rm div}(g) - D \ge 0$.

Now let $f \in \Gamma(X,\cO_X(D))$.  This means ${\rm div}(f) + D\ge
0$, and thus
$$
{\rm div}(g^m f) = m\,{\rm div}(g) + {\rm div}(f) = m\,{\rm div}(g) -
D + {\rm div}(f) + D \ge 0
$$
since a sum of effective divisors is effective.  By Proposition 7.2,
we conclude that $g^mf \in \cO_X(X) = R$.  Hence $h = g^m \in R$ has
the desired property.\qed
\medskip

A finitely generated $R$-submodule of $K$ is called a {\it fractional
ideal\/}.  Thus Proposition~9.3 shows that $\Gamma(X,\cO_X(D))$ is a
fractional ideal. 
\bigskip

\ex{9.2} Let $D \ge 0$ be an effective divisor on the affine variety
$X = {\rm Spec}(R)$.  Prove that the fractional ideal
$\Gamma(X,\cO_X(-D))$ is an ordinary ideal (i.e., $\Gamma(X,\cO_X(-D))
\subset R$).  Hint:~As usual, you will use Proposition~7.2.
\bigskip 

We next show that the $R$-module $\Gamma(X,\cO_X(D))$ determines the
entire sheaf $\cO_X(D)$.  Recall that if $g \in R$ is nonzero, the
Zariski open set $X_g$ defined by the nonvanishing of $g$ is ${\rm
Spec}(R_g)$, where $R_g = \{a/g^m \mid a \in R,\ m \ge 0\}$ is the
localization of $R$ at $g$.

\proclaim Proposition 9.4. Let $D$ be a Weil divisor on the normal
affine variety $X = {\rm Spec}(R)$.  If $g \in R$ is nonzero, then
$$
\Gamma(X_g,\cO_X(D)) = \Big\{{f\over g^m} \mid f \in
\Gamma(X,\cO_X(D)),\ m \ge 0\Big\}.\leqno(9.1)
$$

\prf Let $D = \sum_{i = 1}^s a_i D_i$ and write $\{1,\ldots,s\}$ as a
disjoint union $I \cup J$ where $D_i\cap X_g \ne \emptyset$ for $i \in
I$ and $D_j \subset {\bf V}(g)$ for $j \in J$.  

Suppose that $h \in \Gamma(X_g,\cO_X(D))$, so that $({\rm div}(h) +
D)\big|_{X_g} \ge 0$.  Thus ${\rm ord}_{D_i}(h) \ge -a_i$ for $i \in
I$.  Notice that there is no constraint on ${\rm ord}_{D_j}(h)$ for $j
\in J$.  However, we do know that $g$ vanishes on $D_j$ for $j \in I$,
so that ${\rm ord}_{D_j}(g) > 0$.  Then we can pick $m \in \Z$
sufficiently large so that
$$
m\, {\rm ord}_{D_j}(g) + {\rm ord}_{D_j}(h) > 0\quad \hbox{for}\ j \in
J.
$$
Since ${\rm div}(g) \ge 0$, it follows easily that ${\rm div}(g^m h) +
D \ge 0$ on $X$.  Thus $f = g^m h \in \Gamma(X,\cO_X(D))$, and then $h
= f/g^m$ has the desired form.  From here, the proposition follows
easily. \qed
\medskip

Since the open sets $X_g$ for $g \in R -\{0\}$ form a basis for the
Zariski topology of $X = {\rm Spec}(R)$, Proposition~9.4 shows that
the sheaf $\cO_X(D))$ is uniquely determined by its global sections.
\bigskip
\smallskip

\noindent \ul{Coherent Sheaves}.  The right-hand side of (9.1) is the
{\it localization\/} of $\Gamma(X_g,\cO_X(D))$ at $g$.  More
generally, given any finitely generated $R$-module $M$, one can define
its localization $M_g$, and then one gets a unique sheaf
$\widetilde{M}$ on $X = {\rm Spec}(R)$ such that
$$
\Gamma(X_g,\widetilde{M}) = M_g
$$
for any $g \in R-\{0\}$.  See [5, II.5] for details.

For example, if $X = {\rm Spec}(R)$, Theorem 3.1 of \S3 implies
$\widetilde{R} = \cO_X$, and if $D$ is a Weil divisor on $X$,
Proposition 9.4 implies $\widetilde{M} = \cO_X(D)$ for $M =
\Gamma(X,\cO_X(D))$.

This leads to the following general definition.  Suppose that ${\cal
F}$ is a sheaf of $\cO_X$-modules on an arbitrary variety $X$.  Then
${\cal F}$ is {\it coherent\/} if there is an affine open cover
$\{U_i\}_{i\in I}$ of $X$ such that for every $i \in I$, there is a
finitely generated $\cO_X(U_i)$-module $M_i$ such that
$$
{\cal F}\big|_{U_i} = \widetilde{M_i}.
$$
The simplest example of a coherent sheaf is $\cO_X$.  Furthermore, the
above discussion shows that if $D$ is a Weil divisor on a normal
variety $X$, then $\cO_X(D)$ is also coherent.  We will learn more
properties of $\cO_X(D)$ in the next section.
\bigskip
\medskip

\noindent {\bigbf \S10. Invertible Sheaves and Line Bundles}
\bigskip

We next discuss an especially nice class of sheaves.
\bigskip
\smallskip

\noindent \ul{Invertible Sheaves}. Let ${\cal F}$ be a sheaf of
$\cO_X$ modules on a variety $X$.  Then ${\cal F}$ is {\it
invertible\/} if it is locally trivial, i.e., if there is a Zariski
open cover of $\{U_i\}_{i\in I}$ of $X$ such that ${\cal F}\big|_{U_i}
\simeq \cO_X\big|_{U_i}$.  

It follows immediately that $\cO_X$ is invertible.  A more interesting
result is the following characterization of when the sheaves
$\cO_X(D)$ from \S4 are invertible.

\proclaim Theorem 10.1. Let $D$ be a Weil divisor on a normal variety
$X$.  Then $\cO_X(D)$ is an invertible sheaf if and only if $D$ is a
Cartier divisor.

\prf 
First suppose that $D$ is Cartier.  Since invertibility is a local
property and $D$ is locally principal, we may assume that $X = {\rm
Spec}(R)$ is affine and $D = {\rm div}(f)$ for $f \in K$.  Then $D
\sim 0$, so that by Proposition~9.2, we have
$$
\cO_X(D) \simeq \cO_X(0) = \cO_X,
$$ 
where the last equality is by Exercise 9.1.

Going the other way, suppose that $\cO_X(D)$ is invertible.  We need
to prove that $D$ is locally principal.  By restricting to a suitable
affine open subset, we can assume that $X = {\rm Spec}(R)$ and that
$\cO_X \simeq \cO_X(D)$.  Taking global sections, we get an
isomorphism
$$
R \simeq \Gamma(X,\cO_X(D)) \subset K.
$$
Under this isomorphism, suppose that $1 \in R$ maps to $1/g \in
\Gamma(X,\cO_X(D))$.  The proof of Proposition 9.2 shows that if
we set $E = D - {\rm div}(g)$, then $g\Gamma(X,\cO_X(D)) =
\Gamma(X,\cO_X(E))$.  Thus $\Gamma(X,\cO_X(E)) = R$, so that $\cO_X =
\cO_X(E)$.  If we can show that this forces $E = 0$, then $D = {\rm
div}(g)$ will follow, proving that $D$ is locally principal and hence
Cartier.

Thus we may assume $\cO_X = \cO_X(E)$.  Then $1 \in
\Gamma(X,\cO_X(E))$, which implies $E \ge 0$.  If $E \ne 0$, then some
irreducible hyperface $Y$ appears in $E$ with positive coefficient.
Observe that any affine open subset of $X$ which meets $Y$ has the
same property.  By Exercise~10.1 below, we can then assume that ${\rm
div}(h) = Y$ for some $h \in R$.  It follows that ${\rm div}(1/h) + E
\ge 0$, so that $1/h \in \Gamma(X,\cO_X(E)) = \Gamma(X,\cO_X) =
R$. Since $h$ is also in $R$, this implies that $h$ is invertible,
which means ${\rm div}(h) = 0$.  This contradicts ${\rm div}(h) = Y$
and proves $E = 0$, as desired. \qed
\bigskip

\ex{10.1} Let $Y$ be an irreducible hypersurface in a normal variety
$X$.  The goal of this exercise is to find an
affine open subset $U$ and a rational function $h \in \C(X)$ such
that $Y\cap U \ne \emptyset$ and ${\rm div}(h)\big|_U = Y\cap U$.  
\item{a.} Explain why there is $h \in \cO_{X,Y}$ with ${\rm ord}_Y(h)
= 1$.   
\item{b.} Show that $h$ from part a has the following two properties:
\itemitem{$\bullet$} ${\rm div}(h) = Y + \sum_{j=1}^r b_j E_j$, where
the $E_j$ are distinct from $Y$.
\itemitem{$\bullet$} $h$ is defined on a Zariski open set $U'$ such
that $U'\cap Y \ne \emptyset$.
\item{c.} Show that $U'-(E_1\cup\cdots\cup E_r)$ is nonempty and has
nonempty intersection with $Y$.
\item{d.} Now show that the desired affine open subset $U$ exists.
\bigskip

We can also improve Proposition 492 as follows.
\bigskip

\ex{10.2} Let $D$ and $E$ be Cartier divisors on a normal variety $X$.
Then $D \sim E$ if and only if $\cO_X(D) \simeq \cO_X(E)$ as
$\cO_X$-modules.  Hint:~Adapt the argument of Theorem~10.1. 
\bigskip

A deeper result is the following.  See Proposition 6.15 of [5, II.6]
for a proof.

\proclaim Theorem 10.2. Let $X$ be a normal variety.  Then every
invertible sheaf on $X$ is isomorphic to $\cO_X(D)$ for some Cartier
divisor $D$ on $X$.

We remark that invertible sheaves are sometimes called {\it locally
free sheaves of rank one\/}.
\smallskip
\bigskip

\noindent \ul{The Picard Group}. Given invertible sheaves ${\cal
F}$ and ${\cal G}$ on $X$, one easily proves that 
$$
{\cal F}\otimes_{\cO_X}\! {\cal G}\quad \hbox{and}\quad {\cal F}^\vee =
\underline{Hom}_{\cO_X}({\cal F},{\cO_X})\leqno(10.1)
$$
are also invertible. It is also easy to show that the canonical map
${\cal F}\otimes_{\cO_X}\! {\cal F}^\vee \to \cO_X$ induces an
isomorphism
$$
{\cal F}\otimes_{\cO_X}\! {\cal F}^\vee \simeq \cO_X,
$$
which explains the name ``invertible''.  These properties show that
the set of isomorphism classes of invertible sheaves on $X$ has a
natural group structure under tensor product.  We call
$$
{\rm Pic}(X) = \{\hbox{isomorphism classes of invertible sheaves on
$X$}\}.\leqno(10.2) 
$$
the {\it Picard group\/} of $X$,

Since we already defined ${\rm Pic}(X)$ in (8.1) of \S8, we need to
explain why these definitions are equivalent.  We begin with the
following important result, whose proof we omit (see Proposition~6.13
of [5, II.6] for a proof).

\proclaim Theorem 10.3.  If $D$ and $E$ are Cartier divisors on a
normal variety $X$, then there are canonical isomorphisms
$$
\eqalign{\cO_X(D+E) &\simeq \cO_X(D)\otimes_{\cO_X} \cO_X(E)\cr
	\cO_X(-D) &\simeq \cO_X(D)^\vee.\cr}
$$

If we combine Theorem 10.2 and 10.3, we get a surjective homomorphism
${\rm Div}(X) \to {\rm Pic}(X)$, and Exercise 5.2 shows that the
kernel is ${\rm Div}_0(X)$.  We conclude that for normal varieties,
the two definitions coincide.  However, the definition given in (10.2)
is more general, since it makes sense for any variety $X$.

We should also note that one can define ${\rm Pic}(X)$ using sheaf
cohomology.  Here is the basic idea.  Let $X$ be a normal variety (for
simplicity), and consider the exact sequence of sheaves
$$
1 \to \cO_X^* \to \C(X)^* \to \C(X)^*/\cO_X^* \to 1.\leqno(10.3)
$$
In \S8, we mentioned that ${\rm Div}(X) = H^0(X,\C(X)^*/\cO_X^*)$.
Taking sheaf cohomology, (10.3) gives the {\it long exact sequence\/}
$$
0 \to H^0(X,\cO_X^*) \to H^0(X,\C(X)^*) \to H^0(X,\C(X)^*/\cO_X^*) \to
H^1(X,\cO_X^*) \to H^1(X,\C(X)^*) \to
$$
One can show that $H^1(X,\C(X)^*) = 0$, and then the above long exact
sequence reduces to
$$
1 \to \cO_X(X)^* \to C(X)^* \to {\rm Div}(X) \to H^1(X,\cO_X^*) \to 0.
$$
Comparing this to Exercise 8.4, we conclude that $H^1(X,\cO_X^*) =
{\rm Pic}(X)$.  This is the sheaf-theoretic definition of the Picard
group.
\smallskip
\bigskip 

\noindent \ul{Rank One Reflexive Sheaves}. Now suppose that $D$ is a
Weil divisor on a normal variety $X$.  If $D$ is not Cartier, then we
know that $\cO_X(D)$ is not invertible.  So what kind of sheaf is it?  

Given any sheaf ${\cal F}$ of $\cO_X$-modules, we can define ${\cal
F}^\vee$ as in (10.1), and there is a canonical map ${\cal F} \to {\cal
F}^{\vee\vee}$.  Then we say that ${\cal F}$ is {\it reflexive of rank
one\/} if:
\item{$\bullet$} There is a nonempty Zariski open set $U$ such that
${\cal F}\big|_U$ is trivial.
\item{$\bullet$} ${\cal F}$ is torsion-free.
\item{$\bullet$} The map ${\cal F} \to {\cal F}^{\vee\vee}$ is an
isomorphism. 
\vskip0pt
\noindent Any invertible sheaf is reflexive.  Of more interest is the
following result.  A proof can be found in [2, Chapter VII] and [8,
Appendix to \S1].

\proclaim Proposition 10.4. If $D$ is a Weil divisor on a normal
variety $X$, then $\cO_X(D)$ is a reflexive sheaf of rank one.

The dual of a reflexive sheaf of rank one is again reflexive of rank
one, though the tensor product ${\cal F}\otimes_{\cO_X}\!{\cal G}$ of
reflexive sheaves of rank one need not be reflexive of rank one.
However, the double dual
$$
({\cal F}\otimes_{\cO_X}\!{\cal G})^{\vee\vee}\leqno(10.4)
$$
is reflexive of rank one.  Furthermore, if $D$ and $E$ are Weil
divisors on $X$, then
$$
\cO_X(D+E) \simeq (\cO_X(D)\otimes_{\cO_X}\!\cO_X(E))^{\vee\vee}.
$$
One can also show that up to isomorphism, every reflexive sheaf of
rank one on $X$ comes from a Weil divisor on $X$.  It follows that the
class group ${\rm Cl}(X)$ can be regarded as the group of isomorphism
classes of rank one reflexive sheaves under the product (10.4).
Details of all of this can be found int [2, Chapter VII] and [8,
Appendix to \S1].

In most of algebraic geometry, invertible sheaves are more important
than rank one reflexive sheaves.  However, there are situations where
rank one reflexive sheaves occur naturally.  An example is given by
the {\it canonical sheaf\/} of a Cohen-Macaulay variety $X$, which is only
reflexive of rank one (unless the variety is Gorenstein, in which case
the canonical sheaf is invertible).  The canonical
sheaf plays an important role in duality theory.
\smallskip
\bigskip

\noindent \ul{Line Bundles}. A {\it line bundle\/} over a variety
$X$ consists of a map of varieties $\pi : L \to X$ such that $X$ has
an open cover $\{U_i\}_{i\in I}$ with the following two properties:
\item{$\bullet$} For each $i\in I$, here is an isomorphism $f_i :
\pi^{-1}(U_i) \simeq U_i\times \C$ such that $\pi = \pi_1\circ f_i$,
where $\pi_1 : U_i\times \C \to U_i$ is projection on the first factor.
\item{$\bullet$} For each pair $i,j \in I$, then there is $g_{ij} \in
\cO_X(U_i \cap U_j)^*$ such that the composition
$$
f_j\circ f_i^{-1} : (U_i\cap U_j)\times \C \to (U_i\cap U_j)\times \C 
$$
is given by $(x,\lambda) \mapsto (x,g_{ij}(x)\,\lambda)$.
\vskip0pt
\noindent Since the $g_{ij} \in \cO_X(U_i \cap U_j)^*$ are built from
$f_j\circ f_i^{-1}$, it follows easily that they satisfy the {\it
cocycle condition\/}
$$
g_{ik}(x) = g_{ij}(x) g_{jk}(x)\quad \hbox{for}\quad  i,j,k \in I\, x
\in U_i\cap U_j\cap U_k.\leqno(10.5)
$$
The {\it trivial line bundle\/} is given by $\pi_1 : X \times\C \to
X$, where $\pi_1$ is projection on the first factor. 

Given $x \in X$ and a line bundle $\pi : L \to X$, we call $L_x =
\pi^{-1}(x)$ the {\it fiber\/} of $L$ over $x$.  If $x \in U_i$, we
can use $f_i$ to define an isomorphism $L_x \simeq \C$.  If we also
have $x \in U_j$, then we get a different isomorphism $L_x \simeq \C$,
but the two are related by multiplication by $g_{ij}(x)$.  It follows
that $L_x$ has a natural structure as a 1-dimensional vector space,
i.e., a complex line.  Since $L$ is the union of the $L_x$, this
explains the term ``line bundle''.
\smallskip
\bigskip

\noindent \ul{The Sheaf of Sections of a Line Bundle}. Let $\pi : L
\to X$ be a line bundle over $X$.  If $U \subset X$ is Zariski open,
then a {\it section\/} of $L$ over $U$ is a morphism $s : U \to L$
such that $\pi\circ s(x) = x$ for all $x \in U$.  Then set
$$
\Gamma(U,L) = H^0(U,L) = \{\hbox{all sections of $L$ over
$U$}\}.\leqno(10.6)
$$
Since the fibers are vector spaces, we can add sections and mutiply
them by elements of $\cO_X(U)$.  It follows that (10.6) defines a sheaf
of $\cO_X$-modules.  We will denote this sheaf by $\cO_X(L)$.
\bigskip

\ex{10.3} Let $\pi_1 : X \times \C \to X$ be the trivial bundle.  
\item{a.} Show that a section over $U \subset X$ is described 
by $s(x) = (x,f(x))$, $x \in U$, for a unique $f \in \cO_X(U)$.
\item{b.} Show that the sheaf defined by (10.6) is $\cO_X$.
\bigskip

Now let $L$ be any line bundle over $X$.  Since locally $L$ looks like
$U_i\times \C$, Exercise~10.3 shows that locally, the sheaf $\cO_X(L)$
looks like $\cO_{U_i} \simeq \cO_X\big|_{U_i}$.  We have thus proved
the following result.

\proclaim Proposition 10.5. If $L$ is a line bundle over $X$, then
$\cO_X(L)$ is an invertible sheaf.

We can also reverse this process by showing that {\it every\/}
invertible sheaf is the sheaf of sections of some line bundle.  In the
special case when $X$ is normal, we can do this as follows.  Suppose
that ${\cal L}$ is an invertible sheaf.  By Theorem~10.2, ${\cal L}
\simeq \cO_X(D)$ for some Cartier divisor $D$.  Then let
$\{(U_i,f_i)\}_{i\in I}$ be local data for $D$, so that ${\rm
div}(f_i)\big|_{U_i} = D\big|_{U_i}$ for all $i$.

With this set-up, let $g_{ij} = f_i/f_j$, and note that $g_{ij} \in
\cO_X(U_i\cap U_j)^*$ by Exercise~8.3.  Furthermore, it is obvious
that the $g_{ij}$ satisfy cocycle condition (10.5).  We saw in
\S3 how we can construct $X$ from the $U_i$
by gluing $U_i$ and $U_j$ together along $U_i\cap U_j$.  In the same
way, we can glue $U_i\times\C$ and $U_j\times\C$ together by
identifying 
$$
(x,\lambda) \longleftrightarrow (x,g_{ij}(x)\,\lambda),\leqno(10.7)
$$ 
where
$$
\eqalign{(x,\lambda) &\in (U_i\cap U_j)\times \C \subset
U_i\times\C \cr 
(x,g_{ij}(x)\,\lambda) &\in (U_i\cap U_j)\times \C \subset
U_j\times\C.} 
$$
The cocycle condition (10.5) shows that thes identifications satisfy
the compatibility conditions from the subsection ``Gluing Together
Affine Varieties'' in \S3.  It follows that we can glue
together the $U_i\times\C$ to get a variety $L$.  In the same way, the
projections $U_i\times\C \to U_i$ patch together to give a morphism
$\pi : L \to X$.  We will omit the proof of the following
proposition. 

\proclaim Proposition 10.6. $\pi : L \to X$ is a line bundle whose
sheaf of sections is isomorphic to the invertible sheaf ${\cal L}
\simeq \cO_X(D)$ we began with.  

It follows that we have three closely related objects:\ Cartier
divisors, invertible sheaves, and line bundles.  In algebraic
geometry, it is customary (though slightly inaccurate) to use the
terms ``invertible sheaf'' and ``line bundle'' interchangeably.
\smallskip
\bigskip

\noindent \ul{The Zero Divisor of a Section\/}. Suppose that the
sheaf of sections of a line bundle $L$ is the invertible sheaf
$\cO_X(D)$, where $D$ is a Cartier divisor.  Then we can think of a
global section in two very different ways: as a section $s : X \to L$
such that $\pi\circ s = 1_X$, and as a rational function $f \in
\C(X)^*$ such that ${\rm div}(f) + D \ge 0$.  How are these related?  

The easiest way to see the link between these notions of ``global
section'' is to define the {\it zero divisor\/} of a nonzero section
$s : X \to L$.  Given such an $s$, consider an open covering
$\{U_i\}_{i\in I}$ which trivializes the bundle.  Then, using the
restriction of $s$ to $U_i$, we get the composition
$$
s_i : U_i \to \pi^{-1}(U_i) \simeq U_i\times\C \to \C.\leqno(10.8)
$$
This is a morphism, so that $s_i \in \cO_X(U_i)$.  Furthermore, one
checks that $s_i = g_{ij} s_j$ on $U_i\cap U_j$.  Since $g_{ij} \in
\cO_X(U_i\cap U_j)^*$, It follows easily that the divisors ${\rm
div}(s_i)$ on $U_i$ patch together to give a divisor on $X$.  This
divisor is clearly locally principal (it equals ${\rm div}(s_i)$ on
$U_i$).  Thus we get a Cartier divisor
$$
{\rm div}_0(s) \in {\rm Div}(X).
$$
Furthermore, ${\rm div}_0(s) \ge 0$ since each $s_i \in \cO_X(U_i)$.
This relates to the global sections of $\cO_X(D)$ as follows.

\proclaim Theorem 10.7. Let $L$ be the line bundle corresponding to the
invertible sheaf $\cO_X(D)$, and suppose that $s \in \Gamma(X,L)
-\{0\}$ corresponds to $f \in \Gamma(X,\cO_X(D)) -\{0\}$.  Then
$$
{\rm div}_0(s) = {\rm div}(f) + D.
$$

\prf First note that both sides of the equation are effective
divisors.  Given $f \in \C(X)^*$ with ${\rm div}(f) + D \ge 0$, we can
define a section $s$ of $L$ as follows.  We constructed $L$ using the
local data $\{(U_i,f_i)\}_{i\in I}$ for $D$.  Then $D\big|_{U_i} =
{\rm div}(f_i)\big|_{U_i}$, so that
$$
{\rm div}(ff_i)\big|_{U_i} = ({\rm div}(f)+D)\big|_{U_i} \ge 0.
$$ 
If we set $s_i = ff_i$, then Proposition 7.2 shows that $s_i \in
\cO_X(U_i)$.  Furthermore, the construction of $L$ shows that the
sections $U_i \to U_i\times\C$ defined by $x \mapsto (x,s_i(x))$ patch
to give a section $s$ of $L$ over $X$.  (This is part of the proof of
Proposition 10.6.)  Since ${\rm div}_0(s)$ is constructed by patching
together the divisors ${\rm div}(s_i) = {\rm div}(ff_i)$, it follows
easily that ${\rm div}_0(s) = {\rm div}(f) + D$, as claimed.\qed
\medskip

The divisor ${\rm div}_0(s)$ tells us where the section $s$ vanishes.
However, being a divisor, ${\rm div}_0(s)$ records more than just the
hypersurfaces $Y \subset X$ where $s$ is zero---the coefficient of $Y$
in ${\rm div}_0(s)$ also tells us to what order $s$ vanishes on $Y$.
\bigskip

\ex{10.4} Let $L$ be a line bundle over $X$. 
\item{a.} Show that the divisors ${\rm div}_0(s)$ for $s \in
\Gamma(X,L) - \{0\}$ are all linearly equivalent. 
\item{b.} Let $D$ be an effective Cartier divisor on $X$ which is
linearly equivalent to ${\rm div}_0(s)$ for some $s \in\Gamma(X,L) -
\{0\}$.  Prove that $D = {\rm div}_0(t)$ for some $t \in\Gamma(X,L)$
\bigskip

Given a line bundle $L$ over $X$, the set of effective divisors
$$
|L| = \{{\rm div}_0(s) \mid s \in \Gamma(X,L) - \{0\}\}
$$
is called a {\it complete linear system\/}.  This terminology is
justified by part b of the Exercise 10.4.
\bigskip

\ex{10.5} If $L$ is a line bundle on a complete variety $X$.  Prove
that $|L|$ can be identified with the projective space
$\P(\Gamma(X,L))$.  Hint:~Exercise~3.4 will be useful.
\bigskip

Finally, we discuss the ``quotient'' of two sections of a line bundle.
If $s,t$ are nonzero sections of $L$, then for each $x \in X$, $s(x)$
and $t(x)$ are elements of the one-dimensional vector space $L_x$.
This space doesn't have a canonical basis, so we can't regard $s(x)$
and $t(x)$ as numbers.  But if $t(x) \ne 0$, then the ``quotient''
$s(x)/t(x)$ makes sense:~it the unique number $\lambda$ such that $s(x) =
\lambda t(x)$.  This suggests that $s/t$ should be a rational function
on $X$.
\bigskip

\ex{10.6} Let $s,t$ be nonzero sections of $L$ over $X$, and let
$\{U_i\}_{i\in I}$ be an open cover of $X$ which trivializes $L$.
\item{a.} By working on $U_i$, show that $s/t = s_i/t_i$, where $s_i$
is as in (10.8) and $t_i$ is defined similarly.
\item{b.} Explain why $s_i/t_i = s_j/t_j$ as rational functions on
$U_i\cap U_j$.  
\vskip0pt
\noindent Part b gives a well-defined element of $\C(X)^*$ which we
denote $s/t$.  
\bigskip

\ex{10.7} Suppose that $L$ is the line bundle built from the Cartier
divisor $D$ on $X$.  Let $s,t$ be nonzero sections of $L$ over $X$
which correspond to $f,g \in \Gamma(X,\cO_X(D))$.  Prove that the
rational function $s/t$ of Exercise 10.6 is given by $f/g$.
\smallskip
\bigskip

\noindent \ul{Invertible Sheaves on Projective Space}.  Let
$x_0,\ldots,x_n$ be homogeneous coordinates on $\P^n$.  Recall from
\S2 that $\P^n$ is covered by the open sets $U_i = \P^n-{\bf
V}(x_i)$ and that
$$
\C(\P^n) = \Big\{{f\over g} \mid f,g \in \C[x_0,\ldots,x_n]\
\hbox{homogeneous of equal degree,}\ g \ne 0\Big\}. 
$$

Now let $H = {\bf V}(x_0) \subset \P^n$.  This is clearly a divisor,
and is Cartier since $\P^n$ is smooth.  Our goal is to determine the
global sections of $\cO_{\P^n}(dH)$ for $d > 0$.
\bigskip

\ex{10.8} Show that $\{(U_i,x_0^d/x_i^d)\}_{0\le i\le n}$ is local data
for $dH$.
\bigskip

We can now describe the global sections of $\cO_{\P^n}(dH)$.

\proclaim Proposition 10.8.  If $d > 0$, then the global sections of
$\cO_{\P^n}(dH)$ are
$$
\Gamma(\P^n,\cO_{\P^n}(dH)) = \Big\{{f\over x_0^d} \mid f\ \hbox{\rm
is homogeneous of degree}\ d\Big\}.
$$


\prf
Let $f/g \in \Gamma(\P^n,\cO_{\P^n}(dH))$, where $f$ and $g$ are
relatively prime.  Then ${\rm div}(f/g) + dH \ge 0$.  If we restrict
to $U_i$, then this becomes $({\rm div}(f/g) + {\rm
div}(x_0^d/x_i^d))\big|_{U_i} \ge 0$.  Equivalently,
$$
{\rm div}(f/g\cdot x_0^d/x_i^d)\big|_{U_i} \ge 0,
$$
so that $f/g\cdot x_0^d/x_i^d \in \cO_{\P^n}(U_i)$.  Can can think of
$U_i$ as a copy of $\C^n$ with variables ${x_j\over x_i}$ for $j \ne
i$.  If $f,g$ have degree $m$, then we can write $f/g\cdot
x_0^d/x_i^d$ as
$$
{f\big({x_0\over x_i},\ldots,{x_n\over x_i}\big) \over g\big({x_0\over
x_i},\ldots,{x_n\over x_i}\big)} \cdot \Big({x_0\over
x_i}\Big)^d.\leqno(10.9)
$$
For each $i$, this must be a polynomial in ${x_j\over x_i}$ for $j \ne
i$.  When $i = 0$, the second factor in (5.9) is 1, which means that
the denominator of the first factor must be constant since $f$ and $g$
are relatively prime.  Multiplying $f$ and $g$ by suitable constants,
we can assume $g\big(1,{x_1\over x_0},\ldots,{x_n\over x_0}\big) = 1$,
and then $g = x_0^m$ follows since $g$ is homogeneous of degree $m$.

Furthermore, if we then consider (10.9) with $i \ne 0$, then
$f\big({x_0\over x_i},\ldots,{x_n\over x_i}\big) \big({x_0\over
x_i}\big)^{d-m}$ is a polynomial in ${x_j\over x_i}$ for $j \ne i$.
Since $f$ and $g = x_0^m$ are relatively prime, it follows that $d \ge
m$.  Then multiplying $f$ and $g$ by $x_0^{d-m}$ shows that $f$ and
$g$ have the desired form.

Conversely, we need to show that $f/x_0^d$ lies in
$\Gamma(\P^n,\cO_{\P^n}(dH))$ whenever $f$ is homogeneous of degree
$d$.  This follows easily using the above methods.  We omit the
details.
\qed
\bigskip

\ex{10.9} Show that $\Gamma(\P^n,\cO_{\P^n}(dH)) = \{0\}$ if $d < 0$.
\bigskip

It is customary to write $\cO_{\P^n}(dH)$ as $\cO_{\P^n}(d)$.  Then,
up to isomorphism, the global sections of $\cO_{\P^n}(d)$ form the
vector space of homogeneous polynomials of degree $d$.
\bigskip

\ex{10.10} Show that every invertible sheaf on $\P^n$ is isomorphic to
$\cO_{\P^n}(d)$ for some $d \in \Z$.


\medskip
\bigskip


\noindent{\bigbf References}
\bigskip

\parindent=12pt

\smallskip

\item{1.} M.\ F.\ Atiyah and I.\ G.\ MacDonald, {\sl Introduction to
Commutative Algebra\/}, Addison-Wesley, Reading, MA, 1969.
\smallskip

\item{2.} N.\ Bourbaki, {\sl Alg\`ebre commutative\/}, Hermann, Paris,
1965. English translation, Addison-Wesley, Reading, MA, 1972.
\smallskip

\item{3.} D.\ Cox, J.\ Little and D.\ O'Shea, {\sl Ideals, Varieties,
and Algorithms\/}, Second Edition, Springer-Verlag, New
York-Berlin-Heidelberg, 1997.
\smallskip

\item{4.} D.\ Eisenbud and J.\ Harris, {\sl The Geometry of Schemes\/},
Springer-Verlag, New York-Berlin-Heidelberg, 2000.
\smallskip

\item{5.} R.\ Hartshorne, {\sl Algebraic Geometry\/}, Springer-Verlag,
New York-Berlin-Heidelberg, 1977. 
\smallskip

\item{6.} H.\ Matsumura, {\sl Commutative Ring Theory\/}, Cambridge
Univ.\ Press, Cambridge, 1986.
\smallskip

\item{7.} D.\ Prill, {\it Local classification of quotients of complex
manifolds by discontinuous groups\/}, Duke Math.\ J.\ {\bf 34} (1967),
375--386.
\smallskip

\item{8.} M.\ Reid, {\it Canonical 3-folds\/}, in {\sl Journ\'ees de
g\'eom\'etrie alg\'ebrique d'Angers\/} (A.\ Beauville, ed.), Sijthoff
\&Noordhoff, Alphen aan den Rijn, 1980, 273--310. 
\smallskip

\item{9.} J.-P.\ Serre, {\it G\'eom\'etrie alg\'ebrique et g\'eom\'etrie
analytique\/}, Ann.\ Inst.\ Fourier {\bf 6} (1956), 1--42.
\smallskip

\item{10.} B.\ Sturmfels, {\sl Algorithms in Invariant Theory\/},
Springer-Verlag, New York-Berlin-Heidelberg, 1993.
\smallskip

\item{11.} O.\ Zariski and P.\ Samuel, {\sl Commutative Algebra\/},
Volume I, Van Nostrand, Princeton, NJ, 1958.  Reprint by
Springer-Verlag, New York-Berlin-Heidelberg, 1975.


\end