Standard Artinian algebras and Lefschetz properties: A geometric approach

In this paper we deal with Lefschetz properties for standard Artinian Gorenstein algebras (SAGAs). We will briefly present some geometric methods already used by the authors in collaboration with G.P. Pirola which allowed to prove the validity of some weak and strong Lefschetz properties for complete intersection SAGAs presented by quadrics of codimension 5 and 6. We focus on complete intersection SAGAs presented by quadrics where “special” elements appear. In particular, we are interested in elements of degree 1 and 2 whose associated multiplication map from R1 has not maximal rank and in the information they give about the structure of the starting SAGA or, when the SAGA is a Jacobian ring of a smooth cubic X, about the geometry of X itself and its hessian.