The class of the Prym-Brill-Noether divisor

TL;DR

This paper investigates the geometry of Prym-Brill-Noether divisors and related moduli spaces, computing divisor classes and establishing properties like irreducibility, with implications for the classification of these moduli spaces.

Contribution

It computes key divisor class coefficients for Prym-Brill-Noether loci and a Brill-Noether divisor, and proves the irreducibility of a specific divisor, advancing understanding of Prym moduli space geometry.

Findings

The divisor alR_g^r has explicitly computed class coefficients.

The strongly Brill-Noether divisor in arM_{g-1,2} is irreducible.

The moduli space alR_{14,2} is of general type.

Abstract

For $r\geq 3$ and $g= \frac{r(r+1)}{2}$, we study the Prym-Brill-Noether variety $V^r(C,\eta)$ associated to Prym curves $[C,\eta]$. The locus $\mathcal{R}_g^r$ in $\mathcal{R}_g$ parametrizing Prym curves $(C, \eta)$ with nonempty $V^r(C,\eta)$ is a divisor. We compute some key coefficients of the class $[\overline{\mathcal{R}}_g^r]$ in $\mathrm{Pic}_\mathbb{Q}(\overline{\mathcal{R}}_g)$. Furthermore, we examine a strongly Brill-Noether divisor in $\overline{\mathcal{M}}_{g-1,2}$: we show its irreducibility and compute some of its coefficients in $\mathrm{Pic}_\mathbb{Q}(\overline{\mathcal{M}}_{g-1,2})$. As a consequence of our results, the moduli space $\mathcal{R}_{14,2}$ is of general type.