On the irreducibility of Hessian loci of cubic hypersurfaces

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 $n\le 5$, ${\mathcal{H}}_f$ 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.