The weak Lefschetz property for Artinian Gorenstein algebras of codimension three

TL;DR

This paper investigates when certain Artinian Gorenstein algebras of codimension three, linked to numerical semigroups, exhibit the weak Lefschetz property, especially in cases with small initial degree.

Contribution

It establishes conditions under which these algebras have the weak Lefschetz property based on the initial degree of their defining ideal.

Findings

Algebras have the weak Lefschetz property when initial degree is small.

Connection between numerical semigroup Apéry sets and algebraic properties.

Provides criteria for the weak Lefschetz property in specific algebra classes.

Abstract

We study the weak Lefschetz property of a class of graded Artinian Gorenstein algebras of codimension three associated in a natural way to the Ap\'ery set of a numerical semigroup generated by four natural numbers. We show that these algebras have the weak Lefschetz property whenever the initial degree of their defining ideal is small.