The non-Lefschetz locus of vector bundles of rank 2 over $\mathbb{P}^2$

TL;DR

This paper investigates the structure of the non-Lefschetz locus for the first cohomology module of rank 2 vector bundles over the projective plane, showing it has the expected codimension for general bundles.

Contribution

It establishes that the non-Lefschetz locus for the first cohomology of a general rank 2 vector bundle on ^2 has the expected codimension.

Findings

Non-Lefschetz locus has the expected codimension for general bundles.

Focus on the first cohomology module of rank 2 bundles over ^2.

Provides geometric insight into the Lefschetz properties of vector bundles.

Abstract

A finite length graded $R$-module $M$ has the Weak Lefschetz Property if there is a linear element $\ell$ in $R$ such that the multiplication map $\times\ell: M_i\to M_{i+1}$ has maximal rank. The set of linear forms with this property form a Zariski-open set and its complement is called the non-Lefschetz locus. In this paper we focus on the study of the non-Lefschetz locus for the first cohomology module $H_*^1(\mathbb{P}^2,\mathcal{E})$ of a locally free sheaf $\mathcal{E}$ of rank $2$ over $\mathbb{P}^2$. The main result is to show that this non-Lefschetz locus has the expected codimension under the assumption that $\mathcal{E}$ is general.