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

Abstract

We study the functor ${\mathrm{Def}}_E^k$ 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 ${\mathrm{Def}}_E^k$ and the smoothness of some well-known deformation functors and their associated moduli spaces. As a tool for the investigation of ${\mathrm{Def}}_E^k$, 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.