19 Febbraio, 2016 15:30 oclock

Sezione di Geometria, Algebra e loro applicazioni

Local triviality of infinitesimal deformations for varieties with quotient singularities

Barbara Fantechi, Sissa

Aula seminari terzo piano (Dipartimento di matematica)

Let V be a complex projective variety with at most quotient singularities.

If V is smooth, all infinitesimal deformations are locally trivial; Schlessinger proved in 1968 that the same is true if the singularities are in codimension greater or equal to 3.

Let (S,p) be an (\'etale or formal) germ of a (necessarily isolated) surface quotient singularity; we say that V has transversal singularities of type (S,p) along a locally closed smooth subvariety Y if \'etale locally the pair (V,Y) is isomorphic to (S,p) times Y.

In case all codimension 2 singularities of V are transversal of type A_{n-1} (that is 1/n(1,n-1)) or 1/n(1,1), we give sufficient conditions to guarantee that all infinitesimal deformations are locally trivial.

As an application, we show that \bar M_{g,n}, the variety of n-pointed stable curves of genus g, is rigid (i.e., has no nontrivial infinitesimal deformations) for all but a finite number of values for (g,n).

This is partially joint work with Alex Massarenti.