The weak Lefschetz property for monomial ideals of small type

TL;DR

This paper develops a combinatorial approach to analyze the weak Lefschetz property for Artinian quotients by monomial ideals in three variables, providing classifications for certain types and characteristics.

Contribution

It introduces a novel combinatorial framework linking the weak Lefschetz property to perfect matchings and lattice paths, extending classification results for type two quotients.

Findings

Complete classification of type two quotients with the weak Lefschetz property in characteristic zero.

Results on quotients of type at most two in positive characteristic.

New combinatorial methods connecting algebraic properties to lattice path enumerations.

Abstract

In this work a combinatorial approach towards the weak Lefschetz property is developed that relates this property to enumerations of signed perfect matchings as well as to enumerations of signed families of non-intersecting lattice paths in certain triangular regions. This connection is used to study Artinian quotients by monomial ideals of a three-dimensional polynomial ring. Extending a main result in the recent memoir [Boij, Migliore, Mir\'o-Roig, Nagel, Zanello], we completely classify the quotients of type two that have the weak Lefschetz property in characteristic zero. We also derive results in positive characteristic for quotients whose type is at most two.