Degenerations to secant cubic hypersurfaces and limiting Hodge structure

The secant variety of the Veronese surface is a singular cubic fourfold. Degenerations to this specific cubic fourfold and the associated limiting Hodge structures are key ingredients for Hassett and Laza in studying the moduli space of cubic fourfolds and the period mapping. We generalize some results to the cubic hypersurfaces of secant type. Specifically, we compute the limit mixed Hodge structure for families of smooth cubic hypersurfaces degenerating to the cubic hypersurface of secant type. Using Usui’s partial compactification and the resulting limit mixed Hodge structure, we characterize a local extension of the period map associated with the degenerating family.