Limiting Hodge Structures for Cubic Hypersurfaces of Secant Type

Abstract

We study limiting Hodge structures for deformations of cubic hypersurfaces degenerating to specific singular cubics of secant type. Collino, Hassett and Laza have investigated degenerations of cubic threefolds and fourfolds whose central fibers are respectively the secant varieties of the rational normal quartic and the Veronese surface. The limiting Hodge structure bridges the geometric degeneration of varieties with the Hodge-theoretic degeneration. The Veronese surface belongs to the class of Severi varieties, and the rational normal quartic is a hyperplane section of the Veronese surface. We previously investigated similar degeneration problems for the secant varieties of Severi varieties in higher dimensions. In the present paper, we continue to study the limiting Hodge structures for the hyperplane sections of Severi varieties.