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

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 $H_{\ast }^1({\mathbb{P}}^2,\mathcal{E})$ 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.

References in corpus (1)

• Symmetry, Unimodality, and Lefschetz Properties for Graded Modules