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\begin{document}
\title{Lecture II\\ Genus of 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 $n$. We want to study
\emph{intrinsic} properties of $C$; properties that are independent
of the specific way in which it is expressed as closed sub-variety of
$\bbP^n$. The fundamental invariant associated with $C$ is
its genus $g$. The genus can be defined in many different ways.
The arithmetic genus of $C$ is defined as follows. Consider the
homogeneous co-ordinate ring $R$ of $C$; recall that this is the
quotient of the polynomial ring $k[X_0,\dots,X_n]$ (which is the
homogeneous co-ordinate ring of $\bbP^n$) by the (homogeneous) ideal
generated by the equations defining $C$. The statement that $C$ is of
dimension 1 is equivalent to the statement that there is a linear
subspace $L\cong\bbP^{n-2}$ of $\bbP^{n-2}$ which does not meet $C$;
moreover, the projection away from $L$ gives a morphism $X\to\bbP^1$
which has finite fibres. After a linear change of co-ordinates we may
assume that $L$ is defined by the vanishing of $X_0$ and $X_1$. The
algebraic version of the above statement about $C$ is that the ring
homomorphism $k[X_0,X_1]\to R$ is injective and makes $R$ a finitely
generated graded module over $k[X_0,X_1]$. It follows that the
dimension of $R_m$ is a linear function of $m$ for sufficiently large
$m$. Writing this linear function as $md+1-g$, we note that $d$ is
the rank of $R$ as a module over $k[X_0,X_1]$ which is geometrically
described as the number of points in the general fibre of
$C\to\bbP^1$; this is called the degree of the curve $C$. The number $g$
is called the ``defect'' or arithmetic genus of the curve $C$. Of
course, it needs to be shown that $g$ is non-negative and is independent
of the embedding of $C$ in $\bbP^n$.
Another notion is that of the Todd genus of $C$. Let
$(\bbP^n)^{*}$ be the dual projective space of $\bbP^n$ which we
think of as the space parametrising hyperplanes (linear
$\bbP^{n-1}$'s) in $\bbP^n$. Recall that this is expressed by writing
the correspondence $I$ in $\bbP^n\times(\bbP^n)^{*}$ which is defined
by the equation $\sum_i X_i Y_i=0$ if $X_i$' and $Y_i$'s are homogeneous
co-ordinates in $\bbP^n$ and $(\bbP^n)^{*}$ respectively. Now, consider
the dual variety $D_C$ in $(\bbP^{n})^{*}$, which is the locus of all
hyperplanes that are tangent to $C$. More precisely, we can describe
it as follows. For each point $p$ in $C$, we have a line
$T_p\cong\bbP^1$ which is tangent to $C$ at $p$. Consider the
sub-variety $J$ of $I$ which is the locus of all pairs $(p,H)$ where $p$
lies on $C$ and $H$ is a hyperplane in $\bbP^n$ (and hence a point of
$(\bbP^n)^{*}$) that contains $T_p$. Then $D_C$ is the image of $J$ in
$(\bbP^n)^{*}$. If $k$ is a field of characteristic zero or if we
replace the given embedding of $C$ by a Veronese twist, then $J\to D$
is an isomorphism over an open subset of $D_C$. It follows that $D_C$ is
of dimension $n-1$ and is thus defined by a single homogeneous
polynomial $F$ in the $Y_i$'s. We write the degree of this polynomial as
$2d+2g-2$, where $d$ is the degree of $C$; equivalently the Todd
genus of $C$ is defined as $(\deg(F)/2)+1-d$. Of course, one needs to
show that this is a non-negative integer and independent of the
embedding of $C$ in $\bbP^n$.
Assume that $k$ is a field of characteristic 0. The coefficients of a
given system of equations defining $C$ generate a sub-field $E$ of $k$
which is a finitely generated extension of $\bbQ$, the field of
rational numbers. Any such field can be embedded in $\bbC$, the field
of complex numbers. Choose such an embedding. With all these choices,
we can now take the ``same'' curve over the field of complex numbers.
As seen in other lectures, this gives a compact oriented topological
manifold of (real) dimension 2. Such a manifold can be
visualised as a sphere with $g$ handles for a certain non-negative integer
$g$ which is called the topological genus. As before, one must show that
this definition is independent of the various choices made above.
How does one extend the above definition to fields of non-zero
characteristic. We notice that the abelianised fundamental group of
$C$ is the free abelian group on $2g$ generators. This
motivates the following definition. Let $l$ be a prime number
different from the characteristic of $k$; we also assume that $k$ is
separably closed. Consider the isomorphism classes (over $C$) of finite
smooth morphisms $C'\to C$ which are Galois and of degree $l$. The number
of such classes is $N=(l^{2g}-1)/(l-1)$. In other words the genus of
$C$ is the half of the number of ``digits'' required to express $N$
in base $l$. As before, we need to show that this number is
independent of the $l$ chosen.
Now one way to prove that the various definitions of genus are
independent of the choices made is to show that they are equal! For
example, it is clear the the topological definitions do not depend on
the embedding in $\bbP^n$ and give non-negative integers. On the
other hand it is clear that the algebraic definitions are independent
of any of the choices made in the topological definitions.
Let $N_d$ denote the rational normal curve of degree $d$ in $\bbP^d$;
it is the image of the $d$-tuple Veronese embedding of $\bbP^1$
in $\bbP^d$. Let $d$ be much larger than $g$ and let us choose $g$
disjoint pairs $(P_i,Q_i)$ of distinct points in $N_d$. Let
$L\cong\bbP^{2g-1}$ be the linear span of these $2g$ points and
$A_i\cong\bbP^1$ be the line that joins $P_i$ with $Q_i$. Let
$M\cong\bbP^{g-1}$ be a linear space in $L$ that meets each $A_i$ at
a point $R_i$ that is distinct from $P_i$ and $Q_i$. Let $C_0$ be the
image of $N_d$ in $\bbP^{d-g}$ under projection away from $M$. We
note that (by the non-vanishing of the Vandermonde determinant), $L$
meets $N_d$ precisely (and transversely) in the points $P_i$'s and
$Q_i$'s. So $C_0$ is obtained by identifying (or pinching together)
the pairs of points on $N_d$.
Let us put this more algebraically. The homogeneous co-ordinate ring
$R$ of $N_d$ is the sub-ring of $k[U,V]$ generated by all monomials
$U^{i}V^{d-i}$ of degree $d$; the assignment $T_i\mapsto U^{i}V^{d-i}$
gives a surjective homomorphism $k[T_0,\dots,T_d]\to R$. The line $A_i$
corresponds to a rank-2 quotient $W_i$ of the linear space spanned by
the $T_i$ (the kernel of this map is the ideal defining $A_i$). The
points $P_i$, $Q_i$ and $R_i$ correspond to distinct rank-1 quotients
of $W_i$. It follows that we can think of $R_i$ as the graph of an
isomorphism between the quotients corresponding to $P_i$ and $Q_i$. In
other words, we obtain graded homomorphisms $P_i:R\to k[P]$ and $Q_i:R\to
k[Q]$ and $R_i$ is a graded isomorphism between $k[P]$ and $k[Q]$. Then
the co-ordinate ring $S$ of $C_0$ is the sub-ring of $R$ consisting of
those elements $F$ such that for each $i$ the image of $F$ under $P_i$
and $Q_i$ are identified under the isomorphism given by $R_i$. It is
thus clear that the rank of the graded piece $S_m$ is precisely $g$
less than the rank of the graded piece $R_m$. Since the latter has
rank precisely $md+1$, it follows that $C_0$ is a (singular) curve with
arithmetic genus $g$.
Now, suppose we could show that as we ``deform'' $C_0$ in $\bbP^{d-g}$,
the singularities of $C_0$ will disappear. Moreover, suppose we could
show that such deformations ``reach'' \emph{all} curves of arithmetic
genus $g$ in $\bbP^{d-g}$. These two statements can be proved by the
use of Hilbert schemes (which were introduced in the previous lecture).
Assuming these results let us see why the \emph{topological} genus of the
curves so obtained is also $g$.
How does one deform away the singularity of $C_0$? To understand this let
us see how one can obtain $C_0$ from a smooth oriented compact manifold of
(real) dimension 2. As noted above, such a manifold can be visualised
as a sphere with $g$ handles. Let us choose a loop around each handle
and collapse this loop to a point. We see that we obtain a surface of
the same topological (homeomorphism) type as $C_0$.
To understand the relation between the Todd genus and the topological
genus, we note that a general line $L^*\cong\bbP^1$ in $(\bbP^n)^{*}$
corresponds to a linear subspace $L\cong\bbP^{n-2}$ in $\bbP^n$; each
point of $L$ corresponds to a hyperplane in $\bbP^n$ that contains
$L$. Now, as we mentioned above, a general such $L$ does not meet $C$
and projection away from $L$ can be identified as a morphism $f:X\to L^*$;
the fibre over a point $p$ in $L^*$ being the intersection of the
corresponding hyperplane with $C$. As we saw earlier, $f$ has degree
$d$. Now, the intersection of $L^*$ with the dual variety $D_C$
corresponds precisely to those hyperplanes that meet $C$
tangentially. For a general choice of $L$ the intersection of $D_C$ and
$L$ is transversal; this means that the hyperplane corresponding to
this intersection has exactly $d-1$ points with tangency at one of
these points. (This is called a Lefschetz pencil of hyperplanes for the
curve $C$). Let us choose a triangulation of $L^*\cong\bbP^1$ such
that the points of $L^*\cap D$ are vertices in this triangulation. We
then see that the Euler characteristic of $C$ is
\[
d.\text{Euler characteristic of $\bbP^1$} - \text{cardinality of
$L^*\cap D$}
\]
For a compact oriented manifold of (real) dimension 2, the Euler
characteristic is $2-2g$, thus we see that the Todd genus and
topological genus are the same.
The assertion that the Todd genus is the same as the arithmetic genus
can be seen as the principal assertion of the Riemann-Roch theorem
for curves (which also asserts a bit more!). In particular,
Hirzebruch and Grothendieck were able to generalise much of the above
through their extension of the Riemann-Roch theorem. Numerous
simpler proofs of this theorem have been given since Hirzebruch and
Grothendieck first published their proofs; notably one by V. K. Patodi
using differential geometry and one more recently by M. V. Nori which
is purely algebraic.
Let $X\subset\bbP^n$ be a smooth projective variety and $R$ be its
homogeneous co-ordinate ring. Since this is a finitely generated
graded ring over $k$, the rank of $R_m$ is a polynomial function of
$m$ for all sufficiently large $m$. To express the coefficients of this
polynomial in terms of certain intersection numbers like the degree of
$X$, the degree of the dual variety of $X$ and so on is the consequence
of these generalisations of the Riemann-Roch theorem for curves. At
the same time, such intersection numbers should be computable using
topological properties of $X$.
As an exercise compute the different genuses for a plane curve.
\end{document}