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\begin{document}
\title{Lecture IV\\ Moduli of Curves}
\author{Kapil Hari Paranjape}
\maketitle
We now study the problem of constructing parametric families of curves.
To begin with let us restrict ourselves to the field $\bbC$ of
complex numbers.
As we have seen earlier, given a curve $X$ of degree $d$
in $\bbP^n$, and a general linear subspace $L\cong\bbP^{n-2}$ in
$\bbP^n$, the projection away from $L$ gives a morphism $f:X\to\bbP^1$
for which each fibre has $d$ points except for $2d+2g-2$ points where
the fibre has $d-1$ points; let $S$ denote this finite set of points.
The morphism $f:X\setminus f^{-1}(S)\to\bbP^1\setminus S$ is a covering
space and so is determined by a homomorphism
$\pi_1(\bbP^1\setminus S,p)\to\Perm(f^{-1}(p))$, where $p$ is some choice of
base point and $\Perm(f^{-1}(p))$ is the group of permutations of
this finite set of $d$ points. Since $\pi_1(\bbP^1\setminus S,p)$
is a free group with $|S|-1$ generators, there are only finitely many
such homomorphisms. Conversely, any homomorphism $\pi_1(\bbP^1\setminus
S,p)\to\Perm(T)$ gives a finite (not necessarily connected) cover
whose fibre over $p$ can be identified with $T$. In order that the cover is
connected the image of this group homomorphism should act
transitively on $T$. In order that the cover be similar to the
description above it is necessary and sufficient that for each $s$ in
$S$, the elements of $\pi_1(\bbP^1\setminus S,p)$ corresponding to all
sufficiently small loops around $s$, are mapped to transpositions in
$\Perm(T)$. In summary, each such cover is determined by a map
$l:S\to\Perm(T)$ which lands in transpositions such that $\prod l_s$
is the identity and such that the image generates a transitive subgroup
of $\Perm(T)$. (Exercise: Show that the homomorphism is surjective in
this case).
From the above discussion we only need to note that the space
parametrising such covers is itself a finite cover of the space
parametrising $(2d+2g-2)$-tuples of points in $\bbP^1$. The latter is
parametrised by the the quotient of an open subset of the space of all
homogeneous polynomials of degree $2d+2g-2$ in two variables under the
action of the group of linear changes of co-ordinates in 2 variables.
In other words, the space parametrising such covers is of dimension
$2d+2g-5$. The next question we need to ask is ``how often'' a given
curve of genus $g$ presents itself in this fashion.
For $d$ sufficiently large, any effective divisor $D$ of degree $d$ on
any curve $X$ of genus $g$ occurs as the hyperplane section of an
embedding $X\subset\bbP^{d-g}$; in fact, by the Riemann-Roch theorem
$d>2g$ is good enough. Think of this hyperplane as a point in
$(\bbP^{d-g})^*$. The locus of lines that contain this point can be
identified with a complementary $\bbP^{d-g-1}$. For the general such
$D$ and the general line $L$ containing it we obtain a morphism
$X\to\bbP^1$ as described above; these general conditions thus give
us a (Zariski) open set in a space parametrising pairs $(D,L)$. The
space of effective divisors of degree $d$ has dimension $d$ and thus
the space of such pairs has dimension $2d-g-1$. Thus, for a fixed
curve $X$, the space of all maps $f:X\to\bbP^1$ as described above is
of dimension $2d-g-2$ (since we can ``forget'' the chosen fibre $D$).
Combining the above two paragraphs we obtain Riemann's count for the
moduli of curves of genus $g$; the difference $(2d+2g-5)-(2d-g-2)=3g-3$
is the dimension of the space parametrising curves of genus $g$.
There are a number of difficulties with the above argument. Many of the
arguments can be easily made more formal without losing the geometric
flavour. However, the use of the complex topology and the existence
theorem of Riemann (to ensure that every covering as
described above \emph{comes from} an algebraic curve) are much more
difficult to sort out. In order to carry these arguments out over
other fields one needs the techniques of ``stable'' morphisms between
curves that were introduced and developed by Beauville, Harris, Mumford
and others. Of late, the study of moduli spaces has received fresh
impetus through its connections with ``string theory'' which is an
attempt to provide a unified description of the fundamental
interactions in physics. Theoretical physicists such as Ashoke Sen of
HRI are interested in ``string theory''.
Let us examine some consequences of the Riemann-Roch theorem and our
earlier computations about the genus of plane curves. A smooth curve
in $\bbP^2$ of degree $d$ has genus $(d-1)(d-2)/2$. Now most curves
of degree $d$ are smooth and the space of all homogeneous polynomials
of degree $d$ has dimension $(d+1)(d+2)/2$; moreover, a polynomial
and its transform under a linear change of co-ordinates clearly
describe isomorphic curves. Thus the space of smooth plane curves of
genus $(d-1)(d-2)/2$ has dimension $(d+1)(d+2)/2-9$; put differently,
the space of smooth plane curves of genus $g$ (which only exist when
$g$ is a triangular number) grows only linearly with $g$ whereas the
collection of all curves of genus $g$ grows like $3g$. Thus for all
$g$ sufficiently large (in fact for $g>3$), the general curve of
genus $g$ does not occur as a plane curve even if $g$ is a triangular
number. (Exercise: Find a curve of genus $3$ that does not occur as a
plane curve).
\end{document}