On the Lefschetz locus in Gor(1,n,n,1)

TL;DR

This paper investigates special cubic hypersurfaces with vanishing Hessian, providing rational parametrizations and degree computations, and characterizes the associated Gorenstein algebras satisfying the Strong Lefschetz property for certain dimensions.

Contribution

It introduces explicit families of cubic hypersurfaces with vanishing Hessian, computes their degrees, and describes the corresponding Gorenstein algebras with the Strong Lefschetz property for n ≤ 7.

Findings

Two families exhaust the non-conical cubics with vanishing Hessian for N ≤ 6.

Explicit rational parametrizations of these hypersurfaces.

Description of Gorenstein algebras satisfying the Strong Lefschetz property for n ≤ 7.

Abstract

We study two special families of cubic hypersurfaces with vanishing Hessian in $\mathbb{P}^N$, obtaining rational parametrizations and computing their degree in $\mathbb{P}(S_3)$. For $N \leq 6$, these two families exhaust the locus of cubics with vanishing Hessian that are not cones. As a consequence, via Macaulay-Matlis duality, we obtain a description of the locus in $\mathrm{Gor}(1, n, n, 1)$ corresponding to those algebras that satisfy the Strong Lefschetz property, for $n \leq 7$.