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

We study the functor Def₄^k of infinitesimal deformations of a locally free sheaf E of Oₗ -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 Def₄^k and the smoothness of some well-known deformation functors and their associated moduli spaces. As a tool for the investigation of Def₄^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.