On quotients ofℳ¯g,nby certain subgroups ofSn

Abstract

We investigate when certain quotients of the compactified moduli space of n-pointed genus g curves ℳ ¯ G :=ℳ ¯ g,n /G are of general type or, on the contrary, uniruled, for a fairly broad class of subgroups G of the symmetric group S n which act by permuting the n marked points. We show that the property of being of general type only depends on the transpositions contained in G. Furthermore, in the case that G is the full symmetric group S n or a product S n 1 ×⋯×S n m , we find a narrow transitional band in which ℳ ¯ G changes its behaviour from being of general type to its opposite, i.e. being uniruled, as n increases. As an application we consider the universal difference variety ℳ ¯ g,2n /S n ×S n .