Equivariant versal deformations of semistable curves

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

Abstract

We prove that given any $\mathit{n}$-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 a_ne variety $\mathit{W}$. In other words, we prove that the algebraic stack $\mathfrak{M}_{g,n}$ parametrizing $\mathit{n}$-pointed prestable curves of genus $\mathit{g}$ has an étale neighborhood of [$\mathit{C}$] isomorphic to the quotient stack [$\mathit{W}$/ Aut($\mathit{C}$)].

Abstract

We prove that given any $\mathit{n}$-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 a_ne variety $\mathit{W}$. In other words, we prove that the algebraic stack $\mathfrak{M}_{g,n}$ parametrizing $\mathit{n}$-pointed prestable curves of genus $\mathit{g}$ has an étale neighborhood of [$\mathit{C}$] isomorphic to the quotient stack [$\mathit{W}$/ Aut($\mathit{C}$)].

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