Degenerations to secant cubic hypersurfaces and limiting Hodge structure

TL;DR

This paper investigates the degeneration of Hodge structures in one-parameter degenerations to secant cubic hypersurfaces, extending previous results from cubic fourfolds to more general cases involving Severi varieties.

Contribution

It generalizes the study of limit mixed Hodge structures and period map extensions from secant cubic fourfolds to secant cubic hypersurfaces of Severi varieties.

Findings

Characterization of limit mixed Hodge structures for degenerations to secant cubic hypersurfaces

Extension of the period map using S. Usui's partial compactification

Application to the moduli space of cubic hypersurfaces

Abstract

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.