A Characterization of Secant Varieties of Severi Varieties Among Cubic Hypersurfaces

Baohua Fu,Fyodor L Zak,Yewon Jeong

doi:10.1093/imrn/rnaa223

Abstract

Abstract It is shown that an irreducible cubic hypersurface with nonzero Hessian and smooth singular locus is the secant variety of a Severi variety if and only if its Lie algebra of infinitesimal linear automorphisms admits a nonzero prolongation.

Highlights

Let V m ⊂ PN be an m-dimensional irreducible nondegenerate smooth complex projective variety
The secant variety SV is the closure of the union of lines in PN joining two distinct points of V
It is easy to see that V can be isomorphically projected to a lower-dimensional projective space if and only if SV = PN, which can only occur if m is not too big

Summary

Introduction

Let V m ⊂ PN be an m-dimensional irreducible nondegenerate smooth complex projective variety. The secant varieties of Severi varieties are irreducible cubic hypersurfaces defined by vanishing of the determinant of the corresponding 3 × 3 matrix, and the projective duals of these cubics are naturally isomorphic to the corresponding Severi varieties. Let Y be an irreducible cubic hypersurface Is it true that if aut Y = 0 and Y is not polar defective, Y is the secant variety of a Severi variety?. As suggested in [H], the proof of the Main Theorem splits into two parts: the first one is to show that Y = SY0′ for an irreducible component Y0′ ⊂ Y ′ and the second one is to go through the classification of smooth nondegenerate projective varieties with nonzero prolongation given in [FH1] and [FH2]. It is easy to see that any irreducible hypersurface with vanishing Hessian is dual defective, but the converse is not true, as is shown by the secant varieties of Severi varieties. At the end of the paper we give examples showing that none of the conditions a)–c) of the Main Theorem can be lifted

Dual defective cubic hypersurfaces

Cubic hypersurfaces admitting nonzero prolongations

Now let V