The Kodaira dimension of \(\overline{\mathcal{R}}_{g,2}\) and the Prym-Weierstrass divisor

Abstract

We prove that the moduli space of double covers ramified at two points \(\mathcal{R}_{g,2}\) is of general type for \(g\geq 16\). Furthermore, we consider the Prym-Weierstrass divisor \(\overline{\mathcal{PW}}_g\) in the universal curve \(\overline{\mathcal{CR}}_g\) over the moduli space of Prym curves \(\overline{\mathcal{R}}_g\) and we compute its class in \(\mathrm{Pic}_\mathbb{Q}(\overline{\mathcal{CR}}_g)\). The pushforward to \(\overline{\mathcal{M}}_{g,1}\) of this class was not previously known to be in the effective cone.