  The secant variety of the Veronese surface is a singular cubic fourfold. The
degeneration of Hodge structures of one-parameter degenerations to this secant
cubic fourfold is a key ingredient for B.~Hassett and R.~Laza in studying the
moduli space of cubic fourfolds via the period mapping. We generalize some of
their results to the cubic hypersurface that is the secant variety of a Severi
variety. Specifically, we study the limit mixed Hodge structures associated to
one-parameter degenerations to the secant cubic hypersurface. Considering
S.~Usui's partial compactification of a period domain for Hodge structures of
general weights, we apply the limit mixed Hodge structure to characterize a
local extension of the period map for the corresponding cubic hypersurfaces.
