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

TL;DR

This paper investigates when quotients of the moduli space of pointed genus g curves by certain symmetric group subgroups are of general type, identifying near-optimal conditions and exploring the transition from general type to uniruledness.

Contribution

It establishes conditions under which these quotients are of general type and analyzes the transition zones for different subgroup actions, including applications to universal difference varieties.

Findings

Certain quotients are of general type for specified (g,n)

Transition zones exist where the space changes from general type to uniruled

Application to universal difference varieties with symmetric group actions

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 $\overline{\mathcal{M}}_{g,2n} /S_n \times S_n$.