On the weak Lefschetz property for vector bundles on \(\mathbb{P}^2\) (Q2215776)

Let \(R = \mathbb K[x,y,z]\) be a standard graded polynomial ring, where \(\mathbb K\) is algebraically closed and of characteristic zero. The focus of the paper is on the Weak Lefschetz Property (WLP) for certain finite length graded modules. Recall that a graded module has the WLP if multiplication by some linear form (hence by a general linear form, by semicontinuity) has maximal rank in all degrees. The modules studied in this paper arise as the first cohomology of a rank 2, locally free sheaf \(\mathcal E\) on \(\mathbb P^2\), or equivalently as the cokernel of a map of free modules of ranks \(k+2\) and \(k\) for some \(k\), and the main result says that such modules always have the WLP. The authors also find the degrees where we have injectivity and the degrees where we have surjectivity in the case that \(\mathcal E\) is stable and normalized, and also the case when it is unstable and normalized. Careful use was made of the Buchsbaum-Rim complex. This paper was motivated by a result on complete intersections by \textit{T. Harima} et al. [ibid. 262, No. 1, 99--126 (2003; Zbl 1018.13001)], which was just the case \(k=1\) of this result but introduced the study of the WLP by looking at syzygy sheaves and their cohomology, and applying the Grauert-Mülich theorem. The paper was also motivated by results of \textit{H. Brenner} and \textit{A. Kaid} on this topic [Ill. J. Math. 51, No. 4, 1299--1308 (2007; Zbl 1148.13007)].