Higher Dimensional Brill-Noether Loci and Moduli for Very Ample Line Bundles

In this paper, we study Brill-Noether loci for higher dimensional varieties. Let M be a moduli space of coherent sheaves on X. The Brill-Noether loci of M are the closed subsets ∈ M : h⁰(X,F) ≥ k+1\. When X is a smooth curve and M = PicXᵈ, the space of degree d line bundles on X, the Brill-Noether loci have a natural determinantal scheme structure coming from a map of vector bundles on PicXᵈ, which is the main tool in studying their geometry. In [CMR10], the authors generalize this to the case where X is any variety and M is the moduli space of stable vector bundles on X with fixed invariants but require that Hⁱ(X,E) = 0 for all E ∈ M and i ≥ 2. We generalize these results by showing how to give a natural determinantal scheme structure to the Brill-Noether loci for any X and any M. In doing so we develop the theory of Fitting ideals for a complex which can be used to define natural scheme structures in other cases as well, such as the locus of points where the projective dimension of a coherent sheaf jumps up. As an application of our results, we construct moduli spaces for very ample line bundles on a variety.