Cubic forms having matrix factorizations by Hessian matrices

TL;DR

This paper classifies cubic forms with Hessian matrices that produce matrix factorizations, linking their geometric properties to secant loci and Severi varieties, expanding understanding of their algebraic and geometric structure.

Contribution

It introduces a classification of cubic forms based on Hessian-induced matrix factorizations using XJC-correspondence, connecting algebraic forms to geometric secant and Severi varieties.

Findings

Cubic forms with Hessian matrices induce specific matrix factorizations.

Reduced hypersurfaces satisfy the secant-singularity correspondence.

Singular loci of irreducible forms relate to Severi varieties.

Abstract

Using a part of XJC-correspondence by Pirio and Russo, we classify cubic forms $f$ whose Hessian matrices induce matrix factorizations of themselves. When it defines a reduced hypersurface, it satisfies the "secant-singularity" correspondence, that is, it coincides with the secant locus of its singular locus. In particular, when $f$ is irreducible, its singular locus is either one of four Severi varieties.