Hilbert Scheme of a Pair of Skew Lines on Cubic Hypersurfaces
Abstract
We study an irreducible component H(X) of the Hilbert scheme Hilb2t+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.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.