A Hurwitz divisor on the Moduli of Prym curves

TL;DR

This paper computes specific divisor classes on the moduli space of Prym curves for certain genus and partition conditions, providing new enumerative formulas relevant to algebraic geometry.

Contribution

It introduces explicit calculations of divisor classes on Prym moduli spaces for particular partitions, advancing understanding of their geometric properties.

Findings

Computed first coefficients of divisor classes in the Picard group

Derived enumerative formulas for divisor computations

Enhanced understanding of Prym moduli space geometry

Abstract

For genus $g=2i\geq4$ and the length $g-1$ partition $\mu = (4,2,\ldots,2,-2,\ldots,-2)$ of 0, we compute the first coefficients of the class of $\overline{D}(\mu)$ in $\mathrm{Pic}_\mathbb{Q}(\overline{\mathcal{R}}_g)$, where $D(\mu)$ is the divisor consisting of pairs $[C,\eta]\in \mathcal{R}_g$ with $\eta \cong \mathcal{O}_C(2x_1+x_2+\cdots + x_{i-1}-x_i-\cdots-x_{2i-1})$ for some points $x_1,\ldots, x_{2i-1}$ on $C$. We further provide several enumerative results that will be used for this computation.