The Weak Lefschetz property for quotients by Quadratic Monomials

TL;DR

This paper investigates when Artinian quotients by quadratic monomial ideals fail the Weak Lefschetz Property, extending previous geometric characterizations and exploring new cases of failure.

Contribution

It provides new insights into the failure of the Weak Lefschetz Property for quadratic monomial quotients beyond existing classifications.

Findings

Identifies new cases where the Weak Lefschetz Property fails.

Extends geometric characterization to additional quadratic monomial ideals.

Connects failure cases to classical geometric concepts.

Abstract

In [MMR], Micha\l{}ek--Mir\'o-Roig give a beautiful geometric characterization of Artinian quotients by ideals generated by quadratic or cubic monomials, such that the multiplication map by a general linear form fails to be injective in the first nontrivial degree. Their work was motivated by conjectures of Ilardi and Mezzetti-Mir\'o-Roig-Ottaviani, connecting the failure to Laplace equations and classical results of Togliatti on osculating planes. We study quotients by quadratic monomial ideals, explaining failure of the Weak Lefschetz Property for some cases not covered by [MMR].