Compactification of the Prym map for non cyclic triple coverings

TL;DR

This paper extends the Prym map for non-cyclic triple covers to a proper, finite surjective map of degree 10 from a moduli space of admissible $S_3$-covers of genus 7 to the moduli space of principally polarized abelian surfaces.

Contribution

It constructs a proper extension of the Prym map for non-cyclic triple covers and proves it is finite and surjective with degree 10.

Findings

The Prym map extends to a proper map.

The extended Prym map is finite and surjective.

Degree of the map is 10.

Abstract

In a previous paper, the authors proved that the Prym variety of any non-cyclic etale triple cover of a smooth curve of genus 2 is a Jacobian variety of dimension 2. This gives a map from the moduli space of such covers to the moduli space of Jacobian varieties of dimension 2. We extend this map to a proper map of a certain moduli space of admissible $S_3$-covers of genus 7 to the moduli space of principally polarized abelian surfaces. The main result is that this map is finite surjective of degree 10.