The Weak Lefschetz Property and powers of linear forms in K[x,y,z]

TL;DR

This paper proves that Artinian quotients of three-variable polynomial rings by ideals generated by powers of linear forms possess the Weak Lefschetz property, extending previous results to cases where the syzygy bundle is not semistable.

Contribution

It establishes the Weak Lefschetz property for a broader class of ideals in three variables without requiring semistability of the syzygy bundle.

Findings

Weak Lefschetz property holds for these quotients.

Proof does not rely on syzygy bundle semistability.

Extends previous results to more general cases.

Abstract

We show that an Artinian quotient of K[x, y, z] by an ideal I generated by powers of linear forms has the Weak Lefschetz property. If the syzygy bundle of I is semistable this follows from results of Brenner-Kaid; our proof works without this hypothesis, which typically does not hold.