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 threefold $V$, assuming the presence of a smooth surface $S$ satisfying $C \subset S \subset V$. 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 $\operatorname{Hilb}^{sc} V_4$ of smooth connected curves on a smooth quartic threefold $V_4$ contains infinitely many generically non-reduced irreducible components, which are variations of Mumford's example for $\operatorname{Hilb}^{sc} \mathbb P^3$.