The automorphism group of $\overline{M}_{g,n}$

TL;DR

This paper determines the automorphism groups of the moduli stack and space of stable n-pointed genus g curves, showing they are isomorphic to the symmetric group for most cases, and computes exceptions.

Contribution

It establishes the isomorphism of automorphism groups to the symmetric group for all cases with 2g-2+n≥3 and explicitly computes the remaining cases.

Findings

Automorphism groups are isomorphic to S_n for 2g-2+n≥3.

Explicit automorphism groups computed for remaining cases.

Provides a comprehensive classification of automorphisms for these moduli spaces.

Abstract

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli stack parametrizing Deligne-Mumford stable $n$-pointed genus $g$ curves and let $\overline{M}_{g,n}$ be its coarse moduli space: the Deligne-Mumford compactification of the moduli space of $n$-pointed genus $g$ smooth curves. We prove that the automorphism groups of $\overline{\mathcal{M}}_{g,n}$ and $\overline{M}_{g,n}$ are isomorphic to the symmetric group on $n$ elements $S_{n}$ for any $g,n$ such that $2g-2+n\geq 3$, and compute the remaining cases.