  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.
