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 $S_{n_1}\times \dots \times S_{n_m}$, 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 ${\overline{\mathcal{M}}}_{g,2n}/S_n\times S_n$.

arXiv admin note: text overlap with arXiv:1811.01193

Cited by in corpus (1)

• Birational Geometry of moduli spaces of pointed curves