  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.
