$\overline{\mathcal{R}}_{15}$ is of general type

TL;DR

This paper proves that the moduli space of Prym curves of genus 15 is of general type by constructing a specific divisor and analyzing its properties, extending known results for higher genus cases.

Contribution

It introduces a new divisor on $ar{ ext{R}}_{15}$ as a degeneracy locus and demonstrates its role in establishing the space's general type status.

Findings

The moduli space $ar{ ext{R}}_{15}$ is of general type.

A virtual divisor $ar{ ext{D}}_{15}$ is constructed as a degeneracy locus.

The canonical class $K_{ar{ ext{R}}_{15}}$ is shown to be big.

Abstract

We prove that the moduli space $\overline{\mathcal{R}}_{15}$ of Prym curves of genus $15$ is of general type. To this end we exhibit a virtual divisor $\overline{\mathcal{D}}_{15}$ on $\overline{\mathcal{R}}_{15}$ as the degeneracy locus of a globalized multiplication map of sections of line bundles. We then proceed to show that this locus is indeed of codimension one and calculate its class. Using this class, we can conclude that $K_{\overline{\mathcal{R}}_{15}}$ is big. This complements a 2010 result of Farkas and Ludwig: now the spaces $\overline{\mathcal{R}}_g$ are known to be of general type for $g \geq 14$.