On the Weak Lefschetz Property for Vector Bundles on \mathbb P²

2018/03/27 by Gioia Failla, Failla, Gioia, Zachary Flores +3
Mathematics · #Commutative Algebra and Its Applications #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.1803.10337

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 = ⊕_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 × L : M_j → 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 P² then the first cohomology module of \mathcal E, H¹_*(\mathbb P², \mathcal E), has the Weak Lefschetz Property.

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