Hilbert Scheme of a Pair of Skew Lines on Cubic Hypersurfaces

TL;DR

This work analyzes the Hilbert scheme $H(X)$ of a pair of skew lines on a smooth cubic hypersurface $X\subset \mathbb{P}^n$ (with $n\ge 4$), showing $H(X)$ is normal in all dimensions and smooth precisely when $X$ has no higher triple lines. The authors provide a second proof of smoothness for cubic threefolds and give a dimension-counting argument showing that a general cubic lacks higher triple lines, thereby ensuring smoothness in dimensions $\dim X\ge 3$ for generic cases. Central to the approach is a two-step Hilbert–Chow factorization via blow-ups: first along the diagonal of the Fano variety $F$ of lines, then along the strict transform of the incidental locus $\tilde D$, with the singularities controlled by a set-theoretic intersection $\text{Bl}_{\Delta_F}(F\times F)\cap \tilde D$. The degeneracy locus—the higher triple lines—governs whether the second blow-up introduces singularities; in particular, the case $\dim X=3$ recovers smoothness unconditionally, while $\dim X\ge 4$ yields smoothness iff there are no higher triple lines. The results connect to Voisin’s map, Eckardt points, and incidence geometry of lines, and establish a clear modular interpretation for singularities via type (II) and (IV) subschemes and their relation to ambient $P^3$-spaces.

Abstract

We study an irreducible component H(X) of the Hilbert scheme Hilb^{2t+2}(X) of a smooth cubic hypersurface X containing two disjoint lines. For cubic threefolds, H(X) is always smooth, as shown in arXiv:2010.11622. We provide a second proof and generalize this result to higher dimensions. Specifically, for cubic hypersurfaces of dimension at least four, we show H(X) is normal, and it is smooth if and only if X lacks certain "higher triple lines." We characterize H(X) using the Hilbert-Chow morphism and describe its singularities when X is special.

Figures (6)

• Figure 1: $F^{[2]}$ as an "intermediate" Hilbert scheme of a pair of skew lines
• Figure 2: $\mathbb P^3$-section of cubic fourfold along a higher triple line
• Figure 3: Set-theoretical Picture of $\textup{Bl}_{\Delta_F}(F\times F)\cap \tilde{D}$
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Theorems & Definitions (86)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Theorem 1.6
• Proposition 1.7
• Proposition 1.8
• Proposition 1.10
• Definition 2.1