Deformations of morphisms of locally free sheaves. An application to Brill-Noether theory

Elena Martinengo (Università di Torino)

Tuesday 22nd April 15:00-16:00 Maths 311B

Abstract

In a work with Donatella Iacono, we study infinitesimal deformations of morphisms between two locally free sheaves of O_X modules F and G over a smooth variety X. We define a Cech semicosimplicial differential graded Lie algebra and prove that the Thom Whitney dgla associated to it controls the deformations. Unfortunately the dgla defined in this way is quite complicated, but it works over any algebraically closed field of characteristic zero. Moreover its cohomology groups fit into an exact sequence, from which we obtain much information about the tangent and the obstructions spaces of the deformation problem. As an application, we analyse the infinitesimal deformations of a locally free sheaf E over X together with a linear subspace U of its global sections. We deduce some information on the Brill-Noether loci of E over X, such as the description of the tangent cone at some singular points, of the tangent space at some smooth ones, and we establish a link between the smoothness of the functor Def_(E,U) and the injectivity of the Petri map.

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