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} = *Z*_{0}(*f* )- *Z*_{}(*f* ), where *Z*_{0}(*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(*n*_{i}[*P*_{i}]) = *n*_{i}. The *Riemann-Roch theorem* states that if *L* is any line bundle on *X*,

dim_{k}(*X*, *L*) = deg(*D*) + 1 - *g* + dim_{k}(*X*, *L*)

= deg(*D*) + 1 - *g* + dim_{k}(*X*, *L*^{-1})

Here
(*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 *p*_{0} on the curve we have a natural morphism
: (*X*)(*X*) sending an effective divisor *D* to the class
of
*D* - *d*^{ . }*p*_{0}. 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*.