The birational geometry of $\overline{\mathcal{R}}_{g,2}$ and Prym-canonical divisorial strata

TL;DR

This paper investigates the birational geometry of the moduli space of Prym curves, proving it is uniruled for low genus and of general type for high genus, and computes classes of certain divisorial strata.

Contribution

It establishes the birational type of the moduli space of double covers for various genera and computes classes of Prym-canonical divisorial strata.

Findings

Uniruled for 3 ≤ g ≤ 6

General type for g ≥ 16

Computed classes of divisorial strata in the moduli space

Abstract

We prove that the moduli space of double covers ramified at two points $\mathcal{R}_{g,2}$ is uniruled for $3\leq g\leq 6$ and of general type for $g\geq 16$. Furthermore, we consider Prym-canonical divisorial strata in the moduli space $\overline{\mathcal{C}^n\mathcal{R}}_g$ parametrizing $n$-pointed Prym curves, and we compute their classes in $\mathrm{Pic}_\mathbb{Q}(\overline{\mathcal{C}^n\mathcal{R}}_g)$.