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.