Pseudoreflections on Prym Varieties

TL;DR

The paper characterizes Prym varieties with pseudoreflections in moduli spaces for genus g≥5, revealing three explicit families, contrasting with Jacobian varieties where such loci are empty beyond genus 3.

Contribution

It explicitly describes the loci of Prym varieties with pseudoreflections as three non-empty irreducible families for g≥5, highlighting differences from Jacobian varieties.

Findings

Loci of Prym varieties with pseudoreflections form three explicit families for g≥5.

These loci are non-empty and irreducible.

In genus 6, intermediate Jacobians of cubic threefolds with Eckardt points exemplify such Prym varieties.

Abstract

We show that for every g greater or equal than 5, the locus of Prym varieties in the moduli space of principally polarized abelian varieties of dimension g-1 that possess a pseudoreflection of geometric origin is the union of three different non-empty explicit irreducible families. This is in stark contrast to the loci of Jacobian varieties that possess a pseudoreflection of geometric origin, which is empty for any genus greater than 3. In g=6, a distinguished example of Prym varieties with a pseudoreflection is given by intermediate Jacobians of cubic threefolds that possess an Eckardt point.