The strength of the Weak Lefschetz Property

TL;DR

This paper investigates conditions on Hilbert functions of level artinian algebras that imply the Weak Lefschetz Property, providing partial answers to longstanding questions, especially in codimension 3 Gorenstein algebras.

Contribution

It offers new partial results on when level algebras, particularly in codimension 3, satisfy the Weak Lefschetz Property, including cases with initial degree 2 and small Hilbert functions.

Findings

Affirmative WLP for codimension 3 Gorenstein algebras with initial degree 2

WLP holds for level algebras with relatively small Hilbert functions

Determined the maximum socle degree that guarantees the WLP

Abstract

We study a number of conditions on the Hilbert function of a level artinian algebra which imply the Weak Lefschetz Property (WLP). Possibly the most important open case is whether a codimension 3 SI-sequence forces the WLP for level algebras. In other words, does every codimension 3 Gorenstein algebra have the WLP? We give some new partial answers to this old question: we prove an affirmative answer when the initial degree is 2, or when the Hilbert function is relatively small. Then we give a complete answer to the question of what is the largest socle degree forcing the WLP.