  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.
