Hilbert Scheme of a Pair of Skew Lines on Cubic Hypersurfaces

TL;DR

This paper investigates the geometric structure of the Hilbert scheme of pairs of skew lines on smooth cubic hypersurfaces, generalizing known results for threefolds to higher dimensions and characterizing singularities.

Contribution

It provides a new proof of smoothness for cubic threefolds and extends the analysis to higher dimensions, identifying conditions for smoothness and normality of the Hilbert scheme.

Findings

H(X) is always smooth for cubic threefolds.

H(X) is normal for cubic hypersurfaces of dimension at least four.

H(X) is smooth if and only if the hypersurface has no higher triple lines.

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.