\documentclass[11pt]{amsart}
\usepackage[a4paper,scale=0.8,centering,nomarginpar]{geometry}
\usepackage{setspace}
\onehalfspacing
\newcommand{\bbP}{{\mathbb P}}
\newcommand{\bbG}{{\mathbb G}}
\DeclareMathOperator{\Div}{Div}
\DeclareMathOperator{\divisor}{div}
%\newcommand{\to}{\rightarrow}
\usepackage{hyperref}
\begin{document}

\title{Lecture III\\ Riemann-Roch for Curves}

\author{Kapil Hari Paranjape}
\maketitle

A smooth projective curve $C$ over a field $k$ is by definition a smooth
irreducible projective variety of dimension 1 over $k$.  To begin with
let us think of $C\subset\bbP^n$ for some unspecified $n$.  Let us begin
with a more detailed study of the \emph{arithmetic genus}.

As an example consider a curve $C\subset\bbP^2$ defined as the locus of
vanishing of a homogeneous polynomial $F(X,Y,Z)$ of degree $d$. It is
clear that the homogeneous co-ordinate ring of $C$ is $R=k[X,Y,Z]/(F)$
and so the dimension of $R_m$ for all sufficiently large $m$ is
\[
\binom{m+2}{2} - \binom{m-d+2}{2} =
	md + 1 - \frac{(d-1)(d-2)}{2}
\]
Thus we obtain that the arithmetic genus of a curve of degree $d$ in
$\bbP^2$ is $(d-1)(d-2)/2$. We notice that the dimension of $R_m$ for
small $m$ does not obey this rule. In particular, the dimension of $R_1$
is 3 as soon as $F$ has no linear factor. Understanding this
phenomenon will lead us to ``special'' linear series.

As mentioned in an earlier lecture, all morphisms from a curve
$C\subset\bbP^n$ can be seen as a composition of the Veronese embeddings
and projections. From the definition given above it is clear that the
Veronese embedding does not change the arithmetic genus. What about
projection? By iteration, it is enough to examine what happens when we
project from a single point on the curve. If we prove that the
arithmetic genus is unchanged under an isomorphism induced by such a
projection, then we will have proved that the arithmetic genus of a
curve is an intrinsic invariant.

By a linear change of co-ordinates we can assume that point $P$ we
are projecting from has co-ordinates $(1:0:\dots:0)$. In that case the
projection is seen ring-theoretically as taking the sub-ring $S$ of $R$
generated by the variables $X_1$,\dots,$X_n$. Now, we are also
assuming that the curve in $\bbP^{n-1}$ is isomorphic to our original
curve $C$. If we examine what this means in terms of local co-ordinates
on the curve $C$ at the point $P$ we see that $S_m$ consists of those
$F$ in $R_m$ such that the first $m$ Taylor coefficients of $F/X_0^m$
vanish at the point. Now, we use the smooth-ness of the curve at the
point which implies that there is a linear polynomial (say $X_1$) such
that $X_1/X_0$ is the local co-ordinate at the point $P$; then
$X_0^kX_1^{m-k}$ are linearly independent elements of $R_m$ that do not
lie in $S_m$. It follows that for $m$ sufficiently large this subspace
$S_m$ has co-dimension exactly $m$ in $R_m$. We thus see that the
dimension of $S_m$ is $m(d-1)+1-g$. It follows that the arithmetic genus
is not changed by projection.

Consider a curve $C\subset\bbP^n$ of degree $d$; it is natural to
assume that $C$ ``spans'' $\bbP^n$ so that it does not lie in a
linear subspace of $\bbP^n$. Such a curve is said to be
non-degenerate curve in $\bbP^n$ and we will henceforth restrict our
attention to non-degenerate curves. A linear equation $L$ defines a
sub-variety of $C$ consisting of at most $d$ distinct points.
Given any $n$ points in $\bbP^n$, they lie in a $\bbP^{n-1}$. Hence,
we must have $d\geq n$ for a non-degenerate curve. By applying this
argument to the $m$-th Veronese embedding of the curve for sufficiently
large $m$ we see that $md\geq md-g$ or equivalently $g\geq 0$.

Upon replacing the given embedding of $C$ by a suitable Veronese
twist we can assume that the dimension of $R_m$ is $dm+1-g$ for
\emph{all} $m$; note that $d$ has been increased while ensuring this.
Now, for any $k\leq d-g$ and any $k$ points on the curve, consider
all hyperplanes that contain these $k$ points. This gives a linear
system of hyperplanes of dimension at least $d-k-g$ (it could be
bigger if the $k$ points are linearly dependent); moreover, the
residual intersection (after leaving out the fixed $k$ point locus)
is of degree $d-k$. We generalise this statement below.

A divisor on a curve $C$ is a finite formal expression of the form $\sum_i
n_i P_i$ where $P_i$'s are points of $C$ and $n_i$'s are integers; in
other words, the divisor group $\Div(C)$ is the free abelian group on
points of $C$. A divisor is called \emph{effective} if the $n_i$'s
are positive. A divisor $A$ is said to \emph{contain} a divisor $B$
if the difference $A-B$ is effective. The sum $\sum_i n_i$ is
called the degree of the divisor $\sum_i n_i P_i$.

In an obvious way, each element $F$ of the homogeneous co-ordinate
ring $R$ of $C$ defines a divisor $\divisor(F)$. Since every point of $C$
is an algebraic sub-variety, it is clear that every effective divisor
is contained in a divisor of the type $\divisor(F)$ for some $F$ in $R$. A
divisor $D$ is said to be linearly equivalent to zero if it is of the
form $\divisor(F)-\divisor(G)$ where $F$ and $G$ lie in some $R_m$ (for the same
$m$). Two divisors are said to be linearly equivalent if their difference
is linearly equivalent to zero. In particular, it is clear that two divisors
in a linear system as defined above are linearly equivalent.

Fix a divisor $D=A-B$ where $A$ and $B$ are effective. Further,
replace $C$ by its Veronese embedding so that $A$ is contained in
$\divisor(L)$ for some linear $L$ (i.~e.\ $L$ lies in $R_1$), we can then
write $D$ as $\divisor(L)-B'$ where $B'=B+\divisor(L)-A$ is an effective
divisor. The linear system of hyperplane divisors $\divisor(M)$ that
contain $B'$ is gives a collection of effective divisors that are
linearly equivalent to $D$; moreover, as seen above it has dimension
at least $\deg(L)-\deg(B')-g$. Thus we have proved Riemann's
inequality that the linear system of effective divisors linearly
equivalent to a divisor $D$ is of dimension at least $\deg(D)-g$.
In particular, given a divisor $D$ of degree at least $g$ there
\emph{is} an effective divisor that is linearly equivalent to it.

Now, if we fix an integer $k$, then the $k$-fold Veronese embedding of
$\bbP^n\to\bbP^m$ (where $m=\binom{n+k}{k}-1$) has the property that
the image of any $k$ points of $\bbP^n$ in $\bbP^m$ gives a linearly
independent set. (Hint: Use Vandermonde determinant).

Let us replace $C$ by a Veronese embedding chosen so that
$\dim(R_m)=md+1-g$ and so that any $g$ points on $C$ are linearly
independent. Let $D$ be any effective divisor of degree $d-g$. Since
$\dim(R_1)=d+1-g$, we see that $D$ is contained in a hyperplane
divisor $\divisor(L)$ for some $L$. Moreover, $\divisor(L)-D$ is a linearly
independent set and so we see that the linear system of \emph{all}
divisors linearly equivalent to $D$ is of dimension \emph{precisely}
$\deg(D)-g$; such divisors could be called ``general'' divisors since
we are interested in the \emph{special} divisors for which such an
inequality is strict. What we have just shown is that there is an
upper bound on the degree of all special divisors. Let us note that
if $D$ is an effective divisor of degree less than $g$ then it is
automatically special; ``some things become special just by
existing!''

The Riemann-Roch theorem for curves implies the assertion that every
effective special divisor is contained in a divisor linearly equivalent
to a divisor $K_C$ whose degree is $2g-2$; this divisor (or strictly
speaking its linear equivalence class) is called the \emph{canonical
divisor} of $C$. When $g\geq 2$, there is a morphism $C\to\bbP^{g-1}$
such that the inverse image of any hyperplane is a divisor linearly
equivalent to $K_C$. Special divisors on $C$ can then be identified
with effective divisors whose image lies in a hyperplane in $C$. In
particular, any effective divisor $D$ of degree at most $g-1$ is clearly
special. Given such a divisor $D$ let $\langle D\rangle$ denote the linear span of
$D$. In general, we will have $\dim\langle D\rangle=\deg(D)-1$ but in some cases
$\langle D\rangle$ will be smaller dimensional; in other words it will be even
more special. The Riemann-Roch theorem for such divisors is the very
elegant assertion that the dimension of the linear system $|D|$ of
effective divisors linearly equivalent to $D$ is of dimension
$\deg(D)-\dim\langle D\rangle-1$.

An open problem in the study of smooth projective curves is Green's
conjecture; this attracted the attention of S. Ramanan (a partial answer
and new approach formed the basis of my Ph.~D.\ thesis with him). He
and V. Srinivas also gave a new approach to the problem. C. Voisin
proved the conjecture for ``most'' curves a few years ago but since
the conjecture deals with special divisors (and so special curves),
the conjecture should still attract study from algebraic geometers.
A concise statement of the conjecture is that it gives a precise relation
between an explicit resolution of the co-ordinate ring of the canonical
embedding of a curve and the special divisors (or as seen above linear
dependent collections of points) on it.

\end{document}