Zero cycles on Prym varieties

TL;DR

This paper proves that for Prym varieties from unramified double covers, the embedding of a certain divisor induces an injective push-forward on Chow groups, leading to the conclusion that very general cubic threefolds are stably irrational.

Contribution

It establishes the injectivity of the push-forward homomorphism for Prym varieties from unramified double covers, impacting the understanding of rationality of cubic threefolds.

Findings

Injective push-forward homomorphism for Prym varieties from unramified double covers.

No universal codimension two cycle exists on the product of a very general cubic threefold and its Prym variety.

Very general cubic threefolds are stably irrational.

Abstract

In this text we prove that if an abelian variety $A$ admits an embedding into the Jacobian of a smooth projective curve $C$, and if we consider $\Theta_A$ to be the divisor $\Theta_C\cap A$, where $\Theta_C$ denotes the theta divisor of $J(C)$, then the embedding of $\Theta_A$ into $A$ induces an injective push-forward homomorphism (under certain conditions) at the level of Chow groups. We show that this is the case for every Prym varietiy arising from an unramified double cover of smooth projective curves. As a consequence we prove that there does not exist a universal codimension two cycle on the product of a very general cubic threefold and the Prym variety associated to it. Hence we conclude that a very general cubic threefold is stably irrational.