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

Abstract

We prove that the moduli space of double covers ramified at two points ${\mathcal{R}}_{g,2}$ is uniruled for $3\le g\le 6$ and of general type for $g\ge 16$. 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 ${\mathrm{Pic}}_{\mathbb{Q}}({\overline{{\mathcal{C}}^n\mathcal{R}}}_g)$.

19 pages, improved results, comments are welcome