Prolongations of infinitesimal automorphisms of cubic hypersurfaces with nonzero Hessian

TL;DR

This paper characterizes cubic hypersurfaces with nonzero Hessian that have unusually large automorphism groups, specifically secants of Severi varieties, using prolongations of Lie algebras.

Contribution

It provides a new characterization of secants of Severi varieties among cubic hypersurfaces with nonzero Hessian via prolongations of Lie algebras.

Findings

Secants of Severi varieties have large automorphism groups.

Prolongations of Lie algebras can distinguish these special hypersurfaces.

Characterization applies to smooth cubic hypersurfaces with nonzero Hessian.

Abstract

We study the connected component of the automorphism group of a cubic hypersurface over complex numbers. When the cubic hypersurface has nonzero Hessian, this group is usually small. But there are examples with unusually large automorphism groups: the secants of Severi varieties. Can we characterize them by the property of having unusually large automorphism groups? We study this question from the viewpoint of prolongations of the Lie algebras. Our result characterizes the secants of Severi varieties, among cubic hypersurfaces with nonzero Hessian and smooth singular locus, in terms of prolongations of certain type.