On higher Hessians and the Lefschetz properties

TL;DR

This paper generalizes classical results on hypersurfaces with vanishing Hessian and explores their connection to Lefschetz properties, constructing numerous examples of Artinian Gorenstein algebras that fail these properties.

Contribution

It proves the existence of infinitely many hypersurfaces with vanishing higher Hessians and constructs new Artinian Gorenstein algebras that do not satisfy Lefschetz properties.

Findings

Existence of infinitely many hypersurfaces with vanishing higher Hessians.

Construction of Artinian Gorenstein algebras failing Lefschetz properties.

Identification of cases with unimodal Hilbert vectors not satisfying Weak Lefschetz property.

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 $N\geq 3$, $d \geq 3$ 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.