  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 $[W / {\rm Aut}(C)]$.
