Geometry of Brill-Noether loci on Prym varieties

TL;DR

This paper studies the geometric structure of Brill-Noether loci on Prym varieties, revealing how the locus $V^2$ encodes the covering data and describing its singularities and components.

Contribution

It provides a detailed geometric analysis of the Brill-Noether locus $V^2$ on Prym varieties, including its singular locus and irreducible components, extending prior work on its determinative properties.

Findings

The singular locus of $V^2$ is characterized.

The irreducible components of $V^2$ are described.

$V^2$ uniquely determines the covering.

Abstract

Given the Prym variety of an \'etale double cover one can define analogues of the classical Brill-Noether loci on Jacobians of curves. Recent work by Lahoz and Naranjo shows that the Brill-Noether locus $V^2$ completely determines the covering. In this paper we describe the singular locus and the irreducible components of $V^2$.