Tautological rings of Prym varieties

Abstract

The ring of algebraic cycles modulo algebraic equivalence on an abelian variety is an interesting and mysterious object. When the abelian variety is the Jacobian of a smooth curve, Arnaud Beauville defined a certain subring, called the tautological ring, which has become of great interest to a number of mathematicians. Recently Ben Moonen defined the small and the big tautological rings for Jacobians modulo rational equivalence, both of which surject onto the tautological ring of Beauville.

In this thesis, the notions of tautological rings of Beauville and Moonen are generalized to pairs, consisting of an abelian variety and a subvariety. The tautological ring modulo algebraic equivalence is then studied for the pairs: Prym variety $P$ of a double cover $tilde{C} rightarrow C$ and Abel-Prym curve $psi(tilde{C})$. Generators and certain relations, called ``Polishchuk relations", for the tautological ring of the pair $(P, psi(tilde{C}))$ are determined. Given a complete linear system $g^r_d$ on $C$, Beauville constructed and studied two subvarieties $V_0$ and $V_1$ of $P$, called emph{special subvarieties}. He showed that $V_0$ and $V_1$ have the same class in the cohomology ring of $P$. In this thesis it is shown that in many cases $V_0$ and $V_1$ are, in fact, algebraically equivalent. The class of the union of $V_0$ and $V_1$ turns out to belong to the tautological ring and is expressed in terms of its generators.