On the irreducibility of Hessian loci of cubic hypersurfaces

TL;DR

This work resolves the irreducibility and normality of Hessian varieties ${\mathcal H}_f$ associated with smooth cubic hypersurfaces in ${\mathbb P}^n$ for $n\le 5$: ${\mathcal H}_f$ is normal and irreducible precisely when the cubic $f$ is not of Thom–Sebastiani (TS) type. The authors develop a geometric framework built around the Hessian filtration ${\mathcal D}_k(f)$, the incidence variety $\Gamma_f$, and Adler’s triangle structure, complemented by apolar algebra methods (the SAGAs) to control infinitesimal deformations. They prove a TS-characterisation (Theorem B): $f$ is TS exactly when some ${\mathcal D}_{k+1}(f)$ contains a ${\mathbb P}^{k}$, which coincides with Hessian reducibility, while non-TS forms force the Hessian to be irreducible and generically normal. A key ingredient is Theorem C, describing high-dimensional triangle families and showing that, for $n\le 5$, large triangle configurations force vertices away from $X=V(f)$ unless $f$ is TS. The results extend classic findings (Segre’s cubic surface case) to cubic threefolds and fourfolds, and provide a Hessian-based criterion for TS-type detection, with potential Torelli-type implications for Hessian maps.

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.

Theorems & Definitions (60)

• Proposition 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Proposition 1.5
• proof
• Definition 1.7
• Remark 1.8
• Lemma 1.9
• Lemma 1.10