Prym enumerative geometry and a Hurwitz divisor in $\overline{\mathcal{R}}_{2i}$

TL;DR

This paper computes specific divisor classes in the moduli space of Prym curves, revealing new enumerative geometry results related to ramified covers and Prym structures.

Contribution

It introduces new calculations of divisor classes in Prym moduli spaces and presents novel enumerative results involving ramified covers and Prym line bundles.

Findings

Computed first coefficients of divisor classes in Prym moduli spaces

Established new enumerative results for Prym curves and ramified covers

Connected divisor class calculations with Prym enumerative geometry

Abstract

For $i\geq2$, we compute the first coefficients of the class $[\overline{D}(\mu;3)]$ in the rational Picard group of the moduli of Prym curves $\overline{\mathcal{R}}_{2i}$, where $D(\mu;3)$ is the divisor parametrizing pairs $[C,\eta]$ for which there exists a degree $2i$ map $\pi\colon C\rightarrow \mathbb{P}^1$ having ramification profile $(2,\ldots,2)$ above two points $q_1, q_2$, a triple ramification somewhere else and satisfying $\mathcal{O}_C(\frac{\pi^{*}(q_1)-\pi^{*}(q_2)}{2})\cong \eta$. Furthermore, we provide several new Prym enumerative results related to this situation.