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

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.