On the local structure of the Brill–Noether locus of locally free sheaves on a smooth variety

Abstract

We study the functor \operatorname{Def}_{E}^{k} of infinitesimal deformations of a locally free sheaf E of \mathcal{O}_{X} -modules on a smooth variety X , such that at least k independent sections lift to the deformed sheaf. We deduce some information on the k th Brill–Noether locus of E , such as the description of the tangent cone at some singular points, of the tangent space at some smooth ones and some links between the smoothness of the functor \operatorname{Def}_{E}^{k} and the smoothness of some well-known deformation functors and their associated moduli spaces. As a tool for the investigation of \operatorname{Def}_{E}^{k} , we study infinitesimal deformations of the pairs (E,U) , where U is a linear subspace of sections of E . We generalise many classical results concerning the moduli space of coherent systems to the case where E has any rank and X any dimension. This includes a description of its tangent space and the link between smoothness and the injectivity of the Petri map.