Linear spaces in Hessian loci of cubic hypersurfaces

TL;DR

This paper investigates the structure of Hessian hypersurfaces of smooth cubics, revealing conditions under which their loci contain linear spaces, and characterizes the uniqueness of certain Hessian varieties.

Contribution

It establishes that large Hessian loci imply the cubic is of Thom-Sebastiani type and characterizes the Hessian variety of a general cubic of Waring rank 6.

Findings

Hessian locus dimension larger than expected implies Thom-Sebastiani type cubic.

Unique Hessian variety for smooth cubic threefolds with Waring rank 6.

Hessian of a smooth hypersurface is never a cone.

Abstract

In this paper we will study the Hessian hypersurface associated with a smooth cubic. We prove that the existence of a Hessian locus, associated with a smooth cubic form f, of dimension bigger then the expected one, forces the cubic f to be of Thom-Sebastiani type. Moreover, we will analyze the existence of some projective linear spaces in such Hessian loci and their nature in terms of the Hessian matrix. From this, we show that the only smooth cubic threefold having the same Hessian variety as the one associated with a general cubic form f of Waring Rank 6 is f itself. Finally, we prove that the hessian associated with a smooth hypersurface of any degree and dimension is not a cone.