On the Weak Lefschetz Property for Artinian Gorenstein algebras of codimension three

TL;DR

This paper investigates whether all codimension three graded artinian Gorenstein algebras possess the Weak Lefschetz Property, reducing the problem to compressed cases and providing a complete answer for a specific Hilbert function via geometric analysis.

Contribution

It reduces the Weak Lefschetz Property problem to compressed Gorenstein algebras and completely solves the case for Hilbert function (1,3,6,6,3,1) in all characteristics.

Findings

Confirmed the Weak Lefschetz Property for the specific Hilbert function in all characteristics.

Connected algebraic properties to geometric configurations like Hesse configurations.

Provided a geometric criterion for the property in the open case.

Abstract

We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function (1,3,6,6,3,1), we give a complete answer in every characteristic by translating the problem to one of studying geometric aspects of certain morphisms from $\mathbb P^2$ to $\mathbb P^3$, and Hesse configurations in $\mathbb P^2$.