Hilbert scheme of a pair of skew lines on cubic threefolds

Abstract

Two general lines on a smooth cubic threefold X are disjoint and determine an irreducible component of the Hilbert scheme of X. We prove that this component is smooth and isomorphic to the Hilbert scheme of two points of the Fano varieties of lines of X. We also study its relation to the geometry of lines and singularities on the hyperplane sections of X and its relation to Bridgeland moduli spaces.