The divisors of Prym semicanonical pencils

TL;DR

This paper studies the structure of divisors in the moduli space of double covers of curves with semicanonical pencils, proving irreducibility and computing their classes.

Contribution

It proves the irreducibility of two divisors in the moduli space and calculates their divisor classes in the compactification.

Findings

Both divisors are irreducible.

Divisor classes are explicitly computed.

Results extend Teixidor's arguments to Prym semicanonical pencils.

Abstract

In the moduli space $\mathcal{R}_g$ of double \'etale covers of curves of a fixed genus $g$, the locus of covers of curves with a semicanonical pencil decomposes as the union of two divisors $\mathcal{T}^e_g$ and $\mathcal{T}^o_g$. Adapting arguments of Teixidor for the divisor of curves having a semicanonical pencil, we prove that both divisors are irreducible and compute their divisor classes in the Deligne-Mumford compactification $\overline{\mathcal{R}}_g$.