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A pair of disjoint lines on a smooth cubic threefold determines anirreducible component of the Hilbert scheme. We prove that this component issmooth and isomorphic to the blow-up of the symmetric product of Fano varietiesof lines on the diagonal. We also study its relation to the geometry of linesand singularities on the hyperplane sections and its relation to Bridgelandmoduli spaces.

Hilbert Scheme of a Pair of Skew Lines on Cubic Threefolds