Syzygy bundles and the weak Lefschetz property of almost complete intersections

TL;DR

This paper investigates the weak Lefschetz property for Artinian monomial ideals with four generators in three variables, using syzygy bundles and lozenge tilings to analyze stability, splitting types, and characteristic-dependent conditions.

Contribution

It introduces a novel connection between syzygy bundle stability and lozenge tilings to determine the weak Lefschetz property for specific monomial ideals.

Findings

Characterizes semistability of syzygy bundles via lozenge tilings

Determines generic splitting types for these ideals

Provides criteria for the weak Lefschetz property in different characteristics

Abstract

Deciding the presence of the weak Lefschetz property often is a challenging problem. In this work an in-depth study is carried out in the case of Artinian monomial ideals with four generators in three variables. We use a connection to lozenge tilings to describe semistability of the syzygy bundle of such an ideal, to determine its generic splitting type, and to decide the presence of the weak Lefschetz property. We provide results in both characteristic zero and positive characteristic.