Tautological cycles on Jacobian varieties

Abstract

In this paper we study the algebraic structure of the Tautological ring of a Jacobian: by the use of hard-Lefschetz-primitive classes we construct convenient generators that allow us to list and describe all the possible structures that may occur (the explicit list is given for g ≤ 9 and for a few special curves).

Tautological cycles on tropical Jacobians

“…A celebrated result of Ceresa [Cer83] implies that for a generic curve C, the class of W d is not proportional to the class of Θ modulo algebraic equivalence. Beauville in [Bea04] (see also [Pol05,Mar08,Moo09]) has studied results about algebraic equivalence. We believe that the tautological subring of the ring of tropical cycles modulo algebraic equivalence is an interesting object to study.…”

Smash nilpotence on uniruled 3-folds

“…In [Seb13] it is proved that for one dimensional cycles on a variety dominated by a product of curves, smash equivalence and numerical equivalence coincide. The same result can be deduced from [Mar08] and [Her07], where it is shown that for a smooth projective curve C, for any adequate equivalence relation, [C]…”

On the tautological ring of a Jacobian modulo rational equivalence

“…24 in [8] or to Theorem 4 in [7], if p i alg 0 then p j alg 0 for j ≥ i. We generalize this to the case of rational equivalence in this section, which will be needed in the proof of Theorem 1.…”