LAUSR

Abstract 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 Apery 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 ] and I = ( x a , y b − x b − γ z γ , z c , x a − b + γ y b − β , y b − β z c − 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 − 2 2 ⌋ ; • a ≤ 2 b − c and | a − b | + c − 1 ≤ β ≤ b − 1 ; • one of a , b , c is at most five.