Arnaud Beauville

Algebraic cycles on Jacobian varieties

Let J be the Jacobian of a smooth curve C . We will discuss the ring A(J) of algebraic cycles modulo algebraic equivalence on J , more precisely the smallest subring of A(J) which contains [C] and is stable under the natural operations of A (J) . We will show that this "tautological subring" is generated by the classes of the subvarieties C, C+C, etc. We will discuss the relations between these classes discovered by Polishchuk for a general curve, and by Herbaut for curves with special linear systems.