Zero cycles on Prym varieties

2021/12/30 by Banerjee, Kalyan
#14C15 #14C25 #Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2112.15191

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 Θ_A to be the divisor Θ_C∩ A, where Θ_C denotes the theta divisor of J(C), then the embedding of Θ_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.

Related

• Theta divisors of abelian varieties and push-forward homomorphism at the\n level of Chow groups
• Prym varieties and cubic threefolds over ℤ
• Ordinary and almost ordinary Prym varieties
• A Prym construction for the cohomology of a cubic hypersurface
• Cycle classes on abelian varieties and the geometry of the Abel-Jacobi map