Artinian Gorenstein algebras with Macaulay dual generator with fixed Waring rank

Abstract

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 $F_s:={\ell }_1^d+\cdots +{\ell }_s^d\in \mathbb{K}[X_1,\dots ,X_n]$ 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 $A_{F_s}$.

Article accepted for publication in the Revista Matemática Complutense