On higher Hessians and the Lefschetz properties
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 $N\geq 3$, $d \geq 3$ and $2\leq k < \frac{d}{2}$ there are infinitely many irreducible hypersurfaces $X = V(f)\subset \mathbb{P}^N$, of degree $\operatorname{deg}(f)=d$, not cones and such that their Hessian determinant of order $k$, $\operatorname{hess}^k_f$, 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 firstly in \cite{Wa1} and later in \cite{MW}. As an application we construct for each pair $(N.d) \neq (3,3),(3,4)$ infinitely many standard graded Artinian Gorenstein algebras $A$, of codimension $N+1 \geq 4$ and with socle degree $d \geq 3$ which do not satisfy the Strong Lefschetz property, failing at an arbitrary step $k$ with $2\leq k<\frac{d}{2}$. We also prove that for each pair $(N,d)$, $N \geq 3$ and $d \geq 3$ except $(3,3)$, $(3,4)$, $(3,6)$ and $(4,4)$ there are infinitely many standard graded Artinian Gorenstein algebras of codimension $N+1$, with socle degree $d$, with unimodal Hilbert vectors which do not satisfy the Weak Lefschetz property.