We deal with a generalization of a Theorem of P. Gordan and M. Noether on hypersurfaces with vanishing (first) Hessian. We prove that for any given N ≥ 3, d ≥ 3 and 2 ≤ k < d 2 there are irreducible hypersurfaces X = V (f ) ⊂ P N , of degree deg(f ) = d, not cones and such that their Hessian of order k, hess k f , vanishes identically. The vanishing of higher Hessians is closely related with the Strong (or Weak) Lefschetz property for standard graded Artinian Gorenstein algebra, as pointed out first in [Wa1] and later in [MW]. As an application we construct for each pair (N, d) = (3, 3), (3, 4), standard graded Artinian Gorenstein algebras A, of codimension N + 1 ≥ 4 and with socle degree d ≥ 3 which do not satisfy the Strong Lefschetz property, failing at an arbitrary step k with 2 ≤ k < d 2 . We also prove that for each pair (N, d), N ≥ 3 and d ≥ 3 except (3, 3), (3, 4), (3, 6) and (4, 4) there are standard graded Artinian Gorenstein algebras of codimension N + 1, socle degree d, with unimodal Hilbert vectors and which do not satisfy the Weak Lefschetz property.

Abstract. We prove that for N ≤ 6 an irreducible cubic hypersurface with vanishing hessian in P N is either a cone or a scroll in linear spaces tangent to the dual of the image of the polar map of the hypersurface. We also provide canonical forms and a projective characterization of Special Perazzo Cubic Hypersurfaces, which, a posteriori, exhaust the class of cubic hypersurfaces with vanishing hessian, not cones, for N ≤ 6. Finally we show by pertinent examples the technical difficulties arising for N ≥ 7.

We study the general Jordan type of standard graded Artinian Gorenstein algebras, it is a finer invariant than Weak and Strong Lefschetz properties for those algebras. We prove that their Jordan types are determined by the rank of certain Mixed Hessians. We give a description of the possible Jordan types for algebras of low socle degree and low codimension.

We introduce a family of Artinian Gorenstein algebras, whose combinatorial structure characterizes the ones presented by quadrics. Under certain hypotheses these algebras have non-unimodal Hilbert vector. In particular we provide families of counterexamples to the conjecture that Artinian Gorenstein algebras presented by quadrics should satisfy the weak Lefschetz property.

We introduce a new type of Hessian matrix, that we call Mixed Hessian. The mixed Hessian is used to compute the rank of a multiplication map by a power of a linear form in a standard graded Artinian Gorenstein algebra. In particular we recover the main result of [MW] for identifying Strong Lefschetz elements, generalizing it also for Weak Lefschetz elements. This criterion is also used to give a new proof that Boolean algebras have the Strong Lefschetz Property (SLP). We also construct new examples of Artinian Gorenstein algebras presented by quadrics that does not satisfy the Weak Lefschetz Property (WLP); we construct minimal examples of such algebras and we give bounds, depending on the degree, for their existence. Artinian Gorenstein algebras presented by quadrics were conjectured to satisfy WLP in [MN1,MN2], and in a previous paper we construct the first counter-examples (see [GZ]). *Partially supported by the CAPES postdoctoral fellowship, Proc. BEX 2036/14-2.

We study a generalization of Nagata idealization for level algebras. These algebras are standard graded Artinian algebras whose Macaulay dual generator is given explicitly as a bigraded polynomial of bidegree (1, d). We consider the algebra associated to polynomials of the same type of bidegree (d 1 , d 2 ). We prove that the geometry of the Nagata hypersurface of order e is very similar to the geometry of the original hypersurface. We study the Lefschetz properties for Nagata idealizations of order d 1 , proving that WLP holds if d 1 ≥ d 2 . We give a complete description of the associated algebra in the monomial square free case.

We introduce and study higher order Jacobian ideals, higher order and mixed Hessians, higher order polar maps, and higher order Milnor algebras associated to a reduced projective hypersurface. We relate these higher order objects to some standard graded Artinian Gorenstein algebras, and we study the corresponding Hilbert functions and Lefschetz properties.

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