On higher Jacobians, Laplace equations and Lefschetz properties

• DOI: 10.48550/arxiv.2311.02178
• Publication Date: 2023-01-01
• Authors: Charles Almeida; Aline V. Andrade; Rodrigo Gondim
• Keywords: Algebraically closed field; Codimension; Projective variety; Degree (music); Rank (graph theory)

Let $A$ be a standard graded $\mathbb{K}$-algebra of finite type over an algebraically closed field characteristic zero. We use apolarity to construct, for each degree $k$, projective variety whose osculating defect in $s$ is equivalent the non maximality rank multiplication map power general linear form $\times L^{k-s}: A_s \to A_k$. In Artinian case, this notion corresponds failure Strong Lefschetz property $A$, which allows reobtain some foundational theorems field. It also implies SLP codimension two algebras, known result. The results presented work provide new insights on geometry monomial Togliatti systems, and offer geometric interpretation vanishing higher order Hessians.