Cubic hypersurfaces with positive dual defects

TL;DR

This paper classifies certain cubic hypersurfaces with positive dual defect over complex numbers, revealing their geometric structure and conditions under which they contain large linear subvarieties.

Contribution

It provides a classification of non-conical cubic hypersurfaces with positive dual defect, linking their geometry to secant varieties and linear subvarieties.

Findings

If not a cone, the hypersurface is either the secant of its singular locus or contains a large linear subvariety.

The intersection of the singular locus and a general contact locus is contained in this linear subvariety.

The results characterize the geometric structure of cubic hypersurfaces with positive dual defect.

Abstract

We show that if a cubic hypersurface with positive dual defect over the complex number field is not a cone, then either the hypersurface coincides with the secant variety of the singular locus, or the hypersurface contains a linear subvariety of dimension greater than the dual defect such that the intersection of the singular locus and a general contact locus is contained in the linear subvariety.