Stratifying families of torsion-free sheaves on a smooth surface by their singularities

Abstract

Given a torsion-free sheaf F on P 2 the natural morphism from the versal deformation of F to the product of the versal deformations of the various germs at the singular points (the points where F is not locally free) is formally smooth, under suitable hypothesis (e.g. if F is stable). When studying deformations of such sheaves, a natural approach is thus to start with the local problem, namely deformations of torsion-free k[[ x, y]]-modules. In this context, we define a stratification of the singular locus in the base space of a versal deformation of a torsion-free k[[ x, y]]-module. This is achieved by projecting a “Fitting” stratification of the total space. We show that the various strata are irreducible and we identify the corresponding “generic” singularities. They are of the type m p⊕⋯⊕ m p⊕ m p+1⊕⋯⊕ m p+1 , with m the maximal ideal of k[[ x, y,]]. Having finished the local study, we apply the results to stable torsion-free sheaves on P 2 .