Equivariant versal deformations of semistable curves

Abstract

We prove that given any $n$-pointed prestable curve $C$ of genus $g$ with linearly reductive automorphism group $\mathrm{Aut}(C)$, there exists an $\mathrm{Aut}(C)$-equivariant miniversal deformation of $C$ over an affine variety $W$. In other words, we prove that the algebraic stack ${\mathfrak{M}}_{g,n}$ parametrizing $n$-pointed prestable curves of genus $g$ has an étale neighborhood of $[C]$ isomorphic to the quotient stack $[W/\mathrm{Aut}(C)]$.

21 pages