Higher Rank Brill-Noether Theory on DOUBLE-STRUCK CAPITAL P2

Let M-P2(v) be a moduli space of semistable sheaves on P-2, and let B-k(v) subset of M-P2(v) be the Brill-Noether locus of sheaves E with h(0)(P-2,P- E) >= k. In this paper, we develop the foundational properties of Brill-Noether loci on P-2. Set r = r(E) to be the rank and c(1), c(2) the Chern classes. The Brill-Noether loci have natural determinantal scheme structures and expected dimensions dim B-k(v) = dim MP2(v) - k(k - X(E)). When c(1) > 0, we show that the Brill-Noether locus B-r(v) is nonempty. When c(1) = 1, we show all of the Brill-Noether loci are irreducible and of the expected dimension. We show that when mu = c(1)/r > 1/2 is not an integer and c(2) >> 0, the Brill-Noether loci are reducible and describe distinct irreducible components of both expected and unexpected dimension.