Generic Torelli theorem for Prym varieties of ramified coverings

TL;DR

This paper proves conditions under which the Prym map is generically injective for ramified coverings and shows that very generic Prym varieties of dimension at least 4 are not isogenous to Jacobians.

Contribution

It establishes a generic Torelli theorem for Prym varieties of ramified coverings and explores their isogeny relations to Jacobians.

Findings

Prym map is generically injective under specified conditions.

Very generic Prym varieties of dimension ≥4 are not isogenous to Jacobians.

Provides new insights into the structure of Prym varieties and their moduli.

Abstract

In this paper we prove that the Prym map, from the space of double coverings of a curve of genus g with r branch points to the moduli space of abelian varieties, is generically injective if r>6 and g>1, r=6 and g>2, r=4 and g>4, r=2 and g>5. We also show that a very generic Prym variety of dimension at least 4 is not isogenous to a Jacobian.