Syzygy Bundles on P^2 and the Weak Lefschetz Property

TL;DR

This paper investigates the connection between syzygy bundles on the projective plane and the Weak Lefschetz property of Artinian algebras, establishing criteria based on bundle stability and splitting types.

Contribution

It provides a characterization of when Artinian algebras have the Weak Lefschetz property using syzygy bundle stability and splitting types, including new results for complete and almost complete intersections.

Findings

Complete intersections (n=3) always have the Weak Lefschetz property.

Almost complete intersections (n=4) may lack the Weak Lefschetz property.

Non-semistable syzygy bundles imply the Weak Lefschetz property for almost complete intersections.

Abstract

Let K be an algebraically closed field of characteristic zero and let I=(f_1,...,f_n) be a homogeneous R_+-primary ideal in R:=K[X,Y,Z]. If the corresponding syzygy bundle Syz(f_1,...,f_n) on the projective plane is semistable, we show that the Artinian algebra R/I has the Weak Lefschetz property if and only if the syzygy bundle has a special generic splitting type. As a corollary we get the result of Harima et alt., that every Artinian complete intersection (n=3) has the Weak Lefschetz property. Furthermore, we show that an almost complete intersection (n=4) does not necessarily have the Weak Lefschetz property, answering negatively a question of Migliore and Miro-Roig. We prove that an almost complete intersection has the Weak Lefschetz property if the corresponding syzygy bundle is not semistable.