Artinian Gorenstein algebras with Macaulay dual generator with fixed Waring rank

In this paper, we prove that the Artinian Gorenstein K K -algebra AFₛ A F s of codimension n , socle degree d and Macaulay dual generator Fₛ:= ₁ᵈ + + ₛᵈ ∈ K[X₁, , Xₙ] F s : = ℓ 1 d + ⋯ + ℓ s d ∈ K [ X 1 , ⋯ , X n ] where ₁, ⋯ , 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 Fₛ F s is exactly s . Furthermore, we show that AFₛ A F s is the doubling of a suitable 0-dimensional scheme ZFₛ Z F s in Pⁿ⁻¹ P n - 1 , the so-called tight annihilating scheme of AFₛ A F s , and we compute the minimal free resolution of AFₛ A F s in terms of the minimal free R -resolution of I(ZFₛ) I ( Z F s )