Smoothness Criterion for Hilbert Schemes of Skew Line Pairs on Cubic Hypersurfaces
The pairs of disjoint lines lying on a smooth cubic hypersurface X in projective space, together with their flat limits, form one irreducible component H(X) of the Hilbert scheme of X. A student learns to describe H(X) through the Hilbert-Chow morphism as a double blow-up over the symmetric square of the Fano variety of lines, to classify its points into four subscheme types, and to recognise the higher triple line - a line along which a codimension-two linear section of X acquires a transversal A2 singularity - as the exact obstruction: for dimension at least four, H(X) is smooth precisely when X contains no higher triple line, and is always normal otherwise.
HILBERT SCHEME OF A PAIR OF SKEW LINES ON CUBIC HYPERSURFACES YILONG ZHANG Abstract. We study an
A Hilbert scheme parameterizes all subschemes of a fixed variety with a given Hilbert polynomial; for a smooth cubic hypersurface X in P^n, the pairs of disjoint (skew) lines lying on X and their fla…