On higher Hessians and the Lefschetz properties

Abstract

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<d2 there are irreducible hypersurfaces X=V(f)⊂PN, of degree deg⁡(f)=d, not cones and such that their Hessian of order k, hessfk, 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 [31] and later in [24]. 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<d2. 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.