Brill-Noether theory on the projective plane for bundles with many sections

TL;DR

This paper extends Brill-Noether theory to sheaves on the projective plane, providing bounds, classifications, and geometric properties of loci with many sections, revealing their nonemptiness, irreducibility, and dimension characteristics.

Contribution

It introduces bounds for sections of semistable sheaves on the plane, classifies sheaves near these bounds, and analyzes the geometric properties of associated Brill-Noether loci.

Findings

Upper bounds for h^0(E) in terms of rank and slope.

Classification of sheaves achieving the bounds.

Nonemptiness and irreducibility of Brill-Noether loci.

Abstract

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 the projective plane. In this paper, we give an upper bound for $h^0(E)$ in terms of the rank $r$ and the slope $\mu$ 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 $h^0(E)$ is sufficiently close to the upper bound. We determine the nonemptiness, irreducibility and dimension of the Brill-Noether loci in the moduli spaces of sheaves with $h^0(E)$ in this range. When they are nonempty, these Brill-Noether loci are irreducible though almost always of larger than the expected dimension.