Next: 2 The Grothendieck-Riemann-Roch theorem Up: Algebraic Cycles Previous: Introduction

# 1 Model case of curves

The topic of algebraic cycles has its origin in the theory of divisors on an algebraic curve, or compact Riemann surface. If X is a non-singular projective curve over an algebraically closed field k, a divisor on X is an element of the free abelian group on the points of X; we denote this free abelian group by (X). If f is a rational function on X, we can associate to it its divisor div(f )X = Z0(f )- Z(f ), where Z0(f ) is the set of zeroes of f, and Z(f ) the set of poles of f, both counted with multiplicity. Such a divisor is called a principal divisor; we denote by P(X) the subgroup of (X) consisting of principal divisors, and we define the (divisor) class group (X) = (X)/P(X).

Let (X) denote the group of line bundles (i.e., invertible sheaves) on X. To any meromorphic section of a line bundle L we can associate a divisor in a manner analogous to that for meromorphic functions given above. The divisor associated with a holomorphic section of a line bundle is said to be an effective divisor; this is equivalent to the assertion that all the multiplicities of points occuring in the divisor are non-negative. The ratio of any two mermorphic sections of L is a global meromorphic function. Thus there is a natural homomorphism (X)(X). This map is an isomorphism. By abuse of notation we will denote the divisor class of a line bundle L by L also.

There is a homomorphism deg : (X)Z called the degree homomorphism given by deg(ni[Pi]) = ni. The Riemann-Roch theorem states that if L is any line bundle on X,

dimk(X, L) = deg(D) + 1 - g + dimk(X, L)

= deg(D) + 1 - g + dimk(X, L-1)

Here g is the genus of X and is the invertible sheaf of differential 1-forms. As a consequence one easily obtains the identities g = dimk(X,) and deg() = 2g - 2. Moreover, from the fact that (X, L) is zero if deg(L) < 0 we obtain that

(X, L) = deg(D) + 1 - g

The collection of all effective divisors of a fixed degree d form the smooth projective variety (X) (the d-th symmetric product of X with itself). The kernel (X) of deg : (X)Z is also naturally isomorphic to (the group of k-rational points of) an Abelian variety, the Jacobian variety (X). Fixing a point p0 on the curve we have a natural morphism : (X)(X) sending an effective divisor D to the class of D - d . p0. The Abel-Jacobi theorem (which yields the above isomorphism between (X) and (X)) says that the fibre of through a divisor D precisely consists of all effective divisors in the same divisor class as D. Moreover, from the Riemann-Roch theorem we see that is surjective for d g.

Next: 2 The Grothendieck-Riemann-Roch theorem Up: Algebraic Cycles Previous: Introduction
Kapil Hari Paranjape 2002-11-21