Higher Rank Brill–Noether Theory on ℙ2

Abstract Let $M_{{\mathbb {P}}^{2}}(\textbf {v})$ be a moduli space of semistable sheaves on ${\mathbb {P}}^{2}$, and let $B^{k}(\textbf {v}) \subseteq M_{{\mathbb {P}}^{2}}(\textbf {v})$ be the Brill–Noether locus of sheaves $E$ with $h^{0}({\mathbb {P}}^{2}, E) \geq k$. 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 $\dim B^{k}(\textbf {v}) = \dim M_{{\mathbb {P}}^{2}}(\textbf {v}) - k(k - \chi (E))$. When $c_{1}> 0$, we show that the Brill–Noether locus $B^{r}(\textbf {v})$ 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.