Limiting Hodge Structures for Cubic Hypersurfaces of Secant Type
Abstract
The limiting Hodge structure relates the geometric degeneration of varieties to its Hodge-theoretic behavior. We study degenerations of cubic hypersurfaces whose singular central fibers are secant cubics. The pertinent degenerations of cubic threefolds and fourfolds have been extensively studied for compactifying the associated moduli spaces. In the two cases, the secant cubics are constructed by the rational normal quartic and the Veronese surface, respectively. The Veronese surface is a Severi variety, and the rational normal quartic is a hyperplane section of it. In eariler work, we investigated degenerations to the secant cubics of higher-dimensional Severi varieties. Here we study the limiting Hodge structures for degenerations to their hyperplane sections.
Related
- On the density of rational lines on diagonal cubic hypersurfaces, II
- The Second Gap for Minimal Hypersurfaces in the Unit Sphere with Constant Cubic Trace
- The distribution of semi-integral points on a class of singular cubic hypersurfaces
- Hasse principle for smooth cubic hypersurfaces in the function field case
- Elliptic complements of cubic hypersurfaces
BibTeX
@misc{lyu2026limiting,
title = {{Limiting Hodge Structures for Cubic Hypersurfaces of Secant Type}},
author = {Renjie Lyu and Xuanyu Pan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.11198},
eprint = {2607.11198},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.11198},
}
RIS
TY - UNPB
TI - Limiting Hodge Structures for Cubic Hypersurfaces of Secant Type
AU - Renjie Lyu
AU - Xuanyu Pan
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2607.11198
UR - https://arxiv.org/abs/2607.11198
AB - The limiting Hodge structure relates the geometric degeneration of varieties to its Hodge-theoretic behavior. We study degenerations of cubic hypersurfaces whose singular central fibers are secant cubics. The pertinent degenerations of cubic threefolds and fourfolds have been extensively studied for compactifying the associated moduli spaces. In the two cases, the secant cubics are constructed by the rational normal quartic and the Veronese surface, respectively. The Veronese surface is a Severi variety, and the rational normal quartic is a hyperplane section of it. In eariler work, we investigated degenerations to the secant cubics of higher-dimensional Severi varieties. Here we study the limiting Hodge structures for degenerations to their hyperplane sections.
ER -