Obstructions to deforming curves on a 3-fold, III: Deformations of curves lying on a K3 surface

We study the deformations of a smooth curve C on a smooth projective 3-fold V, assuming the presence of a smooth surface S satisfying C subset of S subset of V. Generalizing a result of Mukai and Nasu, we give a new sufficient condition for a first order infinitesimal deformation of C in V to be primarily obstructed. In particular, when V is Fano and S is K3, we give a sufficient condition for C to be (un) obstructed in V, in terms of (-2)-curves and elliptic curves on S. Applying this result, we prove that the Hilbert scheme Hilb(sc) V-4 of smooth connected curves on a smooth quartic 3-fold V-4 subset of P-4 contains infinitely many generically non-reduced irreducible components, which are variations of Mumford's example for Hilb(sc) P-3.