Brill–Noether Theory on P² for Bundles with Many Sections

The Brill–Noether theory of curves plays a fundamental role in the theory of curves and their moduli and has been intensively studied since the 19th century. In contrast, Brill–Noether theory for higher dimensional varieties is less understood. It is hard to determine when Brill–Noether loci are nonempty and these loci can be reducible and of larger than the expected dimension. Let E be a semistable sheaf on P². In this paper, we give an upper bound βr,μ for h⁰(E) in terms of the rank r and the slope μ of E. We show that the bound is achieved precisely when E is a twist of a Steiner bundle. We classify the sheaves E such that h⁰(E) is within ∣μ(E)∣ + 1 of βr,μ. We determine the nonemptiness, irreducibility and dimension of the Brill–Noether loci in the moduli spaces of sheaves on P² with h⁰(E) in this range. When they are proper subvarieties, these Brill–Noether loci are irreducible though almost always of larger than the expected dimension.