On the irreducibility of Hessian loci of cubic hypersurfaces

TL;DR

This paper investigates the irreducibility of Hessian varieties of smooth cubic hypersurfaces, establishing conditions under which these varieties are irreducible and normal for dimensions up to five.

Contribution

It proves that for dimensions up to five, the Hessian variety is irreducible and normal unless the hypersurface is of Thom-Sebastiani type, extending previous results to higher dimensions.

Findings

Hessian variety is irreducible and normal for n ≤ 5 unless of Thom-Sebastiani type.

Irreducibility is characterized by the inability to separate variables in the defining polynomial.

The study uses singular locus analysis and infinitesimal computations.

Abstract

We study the problem of the irreducibility of the Hessian variety $\mathcal{H}_f$ associated with a smooth cubic hypersurface $V(f)\subset \mathbb{P}^n$. We prove that when $n\leq5$, $\mathcal{H}_f$ is normal and irreducible if and only if $f$ is not of Thom-Sebastiani type, i.e., roughly, one can not separate its variables. This also generalizes a result of Beniamino Segre dealing with the case of cubic surfaces. The geometric approach is based on the study of the singular locus of the Hessian variety and on infinitesimal computations arising from a particular description of these singularities.