On higher Jacobians, Laplace equations and Lefschetz properties

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