Hilbert Scheme of a Pair of Skew Lines on Cubic Hypersurfaces

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.

32 pages, 3 figures, revised, removed section 10