Hilbert Scheme of a Pair of Skew Lines on Cubic Threefolds

Abstract

A pair of disjoint lines on a smooth cubic threefold determines an irreducible component of the Hilbert scheme. We prove that this component is smooth and isomorphic to the blow-up of the symmetric product of Fano varieties of lines on the diagonal. We also study its relation to the geometry of lines and singularities on the hyperplane sections and its relation to Bridgeland moduli spaces.