Geometry of Prym semicanonical pencils and an application to cubic threefolds

TL;DR

This paper explores the geometry of Prym semicanonical pencils within moduli spaces of double covers of curves, revealing differences between divisors and applying findings to enumerate lines on cubic threefolds.

Contribution

It provides a detailed geometric analysis of Prym semicanonical pencils on specific divisors in moduli spaces and connects this to enumerative geometry of cubic threefolds.

Findings

Distinct geometric behaviors of Prym maps on two divisors.

Enumeration of lines on cubic threefolds via analysis of $ au^o_5$.

Rich low-genus geometry of Prym semicanonical pencils.

Abstract

In the moduli space $\mathcal{R}_g$ of double \'etale covers of curves of a fixed genus $g$, the locus formed by covers of curves with a semicanonical pencil consists of two irreducible divisors $\mathcal T^e_g$ and $\mathcal T^o_g$. We study the Prym map on these divisors, which shows significant differences between them and has a rich geometry in the cases of low genus. In particular, the analysis of $\mathcal T^o_5$ has enumerative consequences for lines on cubic threefolds.