Algebraic cycles on Jacobian varieties

Abstract

Let J be the Jacobian of a smooth curve C of genus g, and let A(J) be the ring of algebraic cycles modulo algebraic equivalence on J, tensored with $\mathbb{Q}$. We study in this paper the smallest $\mathbb{Q}$-vector subspace R of A(J) which contains C and is stable under the natural operations of A(J): intersection and Pontryagin products, pull back and push down under multiplication by integers. We prove that this ‘tautological subring’ is generated (over $\mathbb{Q}$) by the classes of the subvarieties $W_1=C,\ W_2=C + C, \dots ,W_{g-1}$. If C admits a morphism of degree d onto $\mathbb{P}^1$, we prove that the last d - 1 classes suffice.