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

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, where $h^0(E)\ge k$. 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 know deformations 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 to the case where $E$ has any rank and $X$ any dimension many classical results concerning the moduli space of coherent systems, like the description of its tangent space and the link between its smoothness and the injectivity of the Petri map.