# Equivariant versal deformations of semistable curves

Alper, Jarod D; Kresch, Andrew (2016). Equivariant versal deformations of semistable curves. Michigan Mathematical Journal, 65(2):227-250.

## Abstract

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

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