Vol. 56, No. 1, 1975

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The local rigidity of the moduli scheme for curves

Robert F. Lax

Vol. 56 (1975), No. 1, 175–178

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).

Mathematical Subject Classification 2000
Primary: 14H10
Secondary: 14D15
Received: 1 November 1973
Revised: 7 May 1974
Published: 1 January 1975
Robert F. Lax