Higher rank Brill-Noether theory on P^2

Abstract

Let $M_{{\mathbb{P}}^2}(v)$ be a moduli space of semistable sheaves on ${\mathbb{P}}^2$, and let $B^k(v)\subseteq M_{{\mathbb{P}}^2}(v)$ be the \textit{Brill-Noether locus} of sheaves $E$ with $h^0({\mathbb{P}}^2,E)\ge k$. In this paper we develop the foundational properties of Brill-Noether loci on ${\mathbb{P}}^2$. 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 $dim{B}^k(v)=dim{M}_{{\mathbb{P}}^2}(v)-k(k-\chi (E))$. 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 $c_2\gg 0$, the Brill-Noether loci are reducible and describe distinct irreducible components of both expected and unexpected dimension.

29 pages, published version. Comments welcome