The weak Lefschetz property of Gorenstein algebras of codimension three associated to the Apéry sets

It has been conjectured that all graded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded algebras A of the Apéry set of M-pure symmetric numerical semigroups generated by four natural numbers. In 2010, Bryant proved that these algebras are graded Artinian Gorenstein algebras of codimension three. In a recent article, Guerrieri showed that if A is not a complete intersection, then A is of form A=R/I with R=K[x,y,z] andI=(xa,yb−xb−γzγ,zc,xa−b+γyb−β,yb−βzc−γ), where 1≤β≤b−1,max⁡{1,b−a+1}≤γ≤min⁡{b−1,c−1} and a≥c≥2. We prove that A has the weak Lefschetz property in the following cases:•max⁡{1,b−a+c−1}≤β≤b−1 and γ≥⌊β−a+b+c−22⌋;•a≤2b−c and |a−b|+c−1≤β≤b−1;•one of a,b,c is at most five.