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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 $ \Div$(X). If f is a rational function on X, we can associate to it its divisor div(f )X = Z0(f )- Z$\scriptstyle \infty$(f ), where Z0(f ) is the set of zeroes of f, and Z$\scriptstyle \infty$(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 $ \Div$(X) consisting of principal divisors, and we define the (divisor) class group $ \Cl$(X) = $ \Div$(X)/P(X).

Let $ \Pic$(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 $ \Pic$(X)$ \to$$ \Cl$(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 : $ \Cl$(X)$ \to$$ \Bbb$Z called the degree homomorphism given by deg($ \sum_{i}^{}$ni[Pi]) = $ \sum_{i}^{}$ni. The Riemann-Roch theorem states that if L is any line bundle on X,

dimk$\displaystyle \HH^{0}_{}$(X, L) = deg(D) + 1 - g + dimk$\displaystyle \HH^{1}_{}$(X, L)

= deg(D) + 1 - g + dimk$\displaystyle \HH^{0}_{}$(X,$\displaystyle \Omega^{1}_{X/k}$ $\displaystyle \otimes$ L-1)

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

$\displaystyle \HH^{0}_{}$(X, L) = deg(D) + 1 - g    $\displaystyle \mbox{if $\deg(D)\geq 2g-1$.}$

The collection of all effective divisors of a fixed degree d form the smooth projective variety $ \Sym^{d}_{}$(X) (the d-th symmetric product of X with itself). The kernel $ \Cl^{0}_{}$(X) of deg : $ \Cl$(X)$ \to$$ \Bbb$Z is also naturally isomorphic to (the group of k-rational points of) an Abelian variety, the Jacobian variety $ \Jac$(X). Fixing a point p0 on the curve we have a natural morphism $ \phi_{d}^{}$ : $ \Sym^{d}_{}$(X)$ \to$$ \Jac$(X) sending an effective divisor D to the class of D - d . p0. The Abel-Jacobi theorem (which yields the above isomorphism between $ \Cl^{0}_{}$(X) and $ \Jac$(X)) says that the fibre of $ \phi_{d}^{}$ 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 $ \phi_{d}^{}$ is surjective for d $ \geq$ g.

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