Artinian Gorenstein algebras with Macaulay dual generator with fixed Waring rank

Abstract In this paper, we prove that the Artinian Gorenstein $$\mathbb {K}$$ K -algebra $$A_{F_s}$$ A F s of codimension n , socle degree d and Macaulay dual generator $$F_s:= \ell _1^d + \dots + \ell _s^d \in \mathbb {K}[X_1, \dots , X_n]$$ F s : = ℓ 1 d + ⋯ + ℓ s d ∈ K [ X 1 , ⋯ , X n ] where $$\ell _1, \cdots , \ell _s$$ ℓ 1 , ⋯ , ℓ s are general linear forms satisfies the strong Lefschetz property (SLP). This result allows us to study whether the Waring rank of $$F_s$$ F s is exactly s . Furthermore, we show that $$A_{F_s}$$ A F s is the doubling of a suitable 0-dimensional scheme $$Z_{F_s}$$ Z F s in $$\mathbb {P}^{n-1}$$ P n - 1 , the so-called tight annihilating scheme of $$A_{F_s}$$ A F s , and we compute the minimal free resolution of $$A_{F_s}$$ A F s in terms of the minimal free R -resolution of $$I(Z_{F_s})$$ I ( Z F s )