Cubic forms having matrix factorizations by Hessian matrices

Abstract

Using a part of XJC-correspondence by Pirio and Russo, we classify cubic forms f 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 f is irreducible, its singular locus is one of the four Severi varieties.