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_$\infty$(f ), where Z₀(f ) is the set of zeroes of f, and Z_$\infty$(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}^{}$n_i[P_i]) = $\sum_{i}^{}$n_i. The Riemann-Roch theorem states that if L is any line bundle on X,

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

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

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

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.