The fibres of the Prym map of \'etale cyclic coverings of degree 7

TL;DR

This paper investigates the structure of Prym varieties from degree 7 cyclic coverings over genus 2 curves, revealing their product structure and describing the fibers of the Prym map in this context.

Contribution

It provides a detailed description of the fibers of the Prym map for degree 7 cyclic coverings, highlighting their relation to Jacobians and automorphisms.

Findings

Prym varieties are products of Jacobians of genus 3 curves.

The polarization type of these Prym varieties is D=(1,1,1,1,1,7).

The fibers of the Prym map are characterized in terms of automorphisms of order 7.

Abstract

We study the Prym varieties arising from \'etale cyclic coverings of degree 7 over a curve of genus 2. These Prym varieties are products of Jacobians JY x JY of genus 3 curves Y with polarization type D=(1,1,1,1,1,7). We describe the fibers of the Prym map between the moduli space of such coverings and the moduli space of abelian sixfolds with polarization type D, admitting an automorphism of order 7.