$\overline{\mathscr{R}}_{15}$ is of general type

Abstract

We prove that the moduli space ${\overline{\mathcal{ℛ}}}_{15}$ of Prym curves of genus $15$ is of general type. To this end we exhibit a virtual divisor ${\overline{\mathcal{D}}}_{15}$ on ${\overline{\mathcal{ℛ}}}_{15}$ as the degeneracy locus of a globalized multiplication map of sections of line bundles. We then proceed to show that this locus is indeed of codimension one and calculate its class. Using this class, we can conclude that $K_{{\overline{\mathcal{ℛ}}}_{15}}$ is big. This complements a 2010 result of Farkas and Ludwig: now the spaces ${\overline{\mathcal{ℛ}}}_g$ are known to be of general type for $g\ge 14$.