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

Given a torsion-free sheaf image on image the natural morphism from the versal deformation of image to the product of the versal deformations of the various germs at the singular points (the points where image is not locally free) is formally smooth, under suitable hypothesis (e.g. if image 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 image, with image the maximal ideal of k[[x,y,]]. Having finished the local study, we apply the results to stable torsion-free sheaves on image.