On the Weak Lefschetz Property for Vector Bundles on $\mathbb P^2$

TL;DR

This paper proves that for any rank 2 locally free sheaf on the projective plane, the associated first cohomology module exhibits the Weak Lefschetz Property, indicating a predictable behavior of multiplication maps by linear forms.

Contribution

It establishes the Weak Lefschetz Property for the first cohomology modules of rank 2 vector bundles on b2, a new result linking vector bundle geometry to algebraic properties.

Findings

First cohomology modules of rank 2 bundles have the Weak Lefschetz Property.

Multiplication by a general linear form has maximal rank on these modules.

Supports the connection between vector bundle geometry and algebraic Lefschetz properties.

Abstract

Let $R=\mathbb K[x,y,z]$ be a standard graded polynomial ring where $\mathbb K$ is an algebraically closed field of characteristic zero. Let $M = \oplus_j M_j$ be a finite length graded $R$-module. We say that $M$ has the Weak Lefschetz Property if there is a homogeneous element $L$ of degree one in $R$ such that the multiplication map $\times L : M_j \rightarrow M_{j+1}$ has maximal rank for every $j$. The main result of this paper is to show that if $\mathcal E$ is a locally free sheaf of rank 2 on $\mathbb P^2$ then the first cohomology module of $\mathcal E$, $H^1_*(\mathbb P^2, \mathcal E)$, has the Weak Lefschetz Property.