具有固定Waring秩的Macaulay对偶生成元的Artinian Gorenstein代数
Artinian Gorenstein algebras with Macaulay dual generator with fixed Waring rank
AI总结：
本文证明了一类具有固定Waring秩的Macaulay对偶生成元的Artinian Gorenstein代数满足强Lefschetz性质，研究了其Waring秩、极小自由分解及线性一般Jordan型。
AI中文摘要：
本文证明，余维数为n、 socle次数为d且Macaulay对偶生成元为F_s:= ℓ₁ᵈ + ⋯ + ℓₛᵈ ∈ 𝕂[X₁, ⋯, Xₙ]（其中ℓ₁, ⋯, ℓₛ为一般线性形式）的Artinian Gorenstein 𝕂-代数A_{F_s}满足强Lefschetz性质（SLP）。该结果使我们可研究F_s的Waring秩是否恰好为s。此外，我们证明A_{F_s}是𝔓ⁿ⁻¹中合适的0维概型Z_{F_s}（即A_{F_s}的紧零化子概型）的加倍，并通过I(Z_{F_s})的极小自由R-分解计算A_{F_s}的极小自由分解。最后，我们确定A_{F_s}的线性一般Jordan型。
英文摘要：
In this paper, we prove that the Artinian Gorenstein $\mathbb{K}$-algebra $A_{F_s}$ of codimension $n$, socle degree $d$ and Macaulay dual generator $F_s := \ell_1^d + \dots + \ell_s^d \in \mathbb{K}[X_1, \dots, X_n]$ where $\ell _1, \cdots , \ell_s$ are general linear forms satisfies the strong Lefschetz property (SLP). This result allows us to study whether the Waring rank of $F_s$ is exactly $s$. Furthermore, we show that $A_{F_s}$ is the doubling of a suitable 0-dimensional scheme $Z_{F_s}$ in $\mathbb{P}^{n-1}$, the so-called tight annihilating scheme of $A_{F_s}$, and we compute the minimal free resolution of $A_{F_s}$ in terms of the minimal free $R$-resolution of $I(Z_{F_s})$. Finally, we determine the linear general Jordan type of $A_{F_s}$.
