A classification of the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms

TL;DR

This paper classifies when certain algebraic structures called almost complete intersections, generated by powers of linear forms, have the Weak Lefschetz property, using Macaulay's inverse system to settle a prior conjecture.

Contribution

It provides a complete classification of the Weak Lefschetz property for these algebras, resolving a previously open conjecture.

Findings

Classification of Weak Lefschetz property for the studied algebras

Use of Macaulay's inverse system to analyze Hilbert series

Resolution of the conjecture by Migliore, Miró-Roig, and Nagel

Abstract

We use Macaulay's inverse system to study the Hilbert series for almost complete intersections generated by uniform powers of general linear forms. This allows us to give a classification of the Weak Lefschetz property for these algebras, settling a conjecture by Migliore, Mir\'o-Roig, and Nagel.