On quotients of \(\overline{\mathcal{M}}_{g,n}\) by certain subgroups of \(S_n\) (Q2675328)

In this article, the property of being of general type or uniruled for the quotient space \(\overline{\mathcal{M}_{g,n}}^G\) of moduli of pointed curves, by a subgroup \(G\) of the permutation group \(S_n\) is studied. The main technique relies on the description of the pullback of the canonical class in terms of \(G\)-invariant effective and ample divisors. As a particular case, the quotient space by the product of symmetric groups is considered.