On quotients of $\overline{\mathcal{M}}_{g,n}$ by certain subgroups of $S_n$

Abstract

We show that certain quotients of the compactified moduli space of $n-$ pointed genus $g$ curves, $\overline{\mathcal{M}}^G:= \overline{\mathcal{M}}_{g,n} / G$, are of general type, for a fairly broad class of subgroups $G$ of the symmetric group $S_n$ which act by permuting the $n$ marked points. The values of $(g,n)$ which we specify in our theorems are near optimal in the sense that, at least in he cases that G is the full symmetric group $S_n$ or a product $S_{n_1}\times \ldots \times S_{n_m}$, there is a relatively narrow transitional zone in which $\overline{\mathcal{M}}^G$ changes its behaviour from being of general type to its opposite, e.g. being uniruled or even unirational. As an application we consider the universal difference variety $\overline{\mathcal{M}}_{g,2n} /S_n \times S_n$.