Let Y be a smooth,
quasi-projective scheme of finite type over an algebraically closed field of
characteristic zero. Let X be the quotient of Y by a finite group of automorphisms.
Assume that the branch locus of Y over X is of codimension at least 3. In this note,
it is shown that X is locally rigid in the following sense: the singular locus of X is
stratifled and, given a point on a stratum, it is shown that there exists a locally
algebraic transverse section to the stratum at the point which is rigid. This result is
then applied to the coarse moduli scheme for curves of genus g, where g > 4 (in
characteristic zero).
