Equivariant versal deformations of semistable curves

TL;DR

This paper constructs an equivariant miniversal deformation space for pointed prestable curves of genus g, showing the local structure of the moduli stack near a given curve is a quotient by its automorphism group.

Contribution

It proves the existence of an Aut(C)-equivariant miniversal deformation for any pointed prestable curve, elucidating the local structure of the moduli stack of curves.

Findings

Existence of an Aut(C)-equivariant miniversal deformation over an affine variety.

The local moduli stack is étale locally a quotient stack by automorphisms.

Provides a framework for understanding automorphism actions on deformation spaces.

Abstract

We prove that given any $n$-pointed prestable curve $C$ of genus $g$ with linearly reductive automorphism group ${\rm Aut}(C)$, there exists an ${\rm Aut}(C)$-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 \'etale neighborhood of $[C]$ isomorphic to the quotient stack $[W / {\rm Aut}(C)]$.