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

Abstract

We study the deformations of a smooth curve [Formula: see text] on a smooth projective [Formula: see text]-fold [Formula: see text], assuming the presence of a smooth surface [Formula: see text] satisfying [Formula: see text]. Generalizing a result of Mukai and Nasu, we give a new sufficient condition for a first order infinitesimal deformation of [Formula: see text] in [Formula: see text] to be primarily obstructed. In particular, when [Formula: see text] is Fano and [Formula: see text] is [Formula: see text], we give a sufficient condition for [Formula: see text] to be (un)obstructed in [Formula: see text], in terms of [Formula: see text]-curves and elliptic curves on [Formula: see text]. Applying this result, we prove that the Hilbert scheme [Formula: see text] of smooth connected curves on a smooth quartic [Formula: see text]-fold [Formula: see text] contains infinitely many generically non-reduced irreducible components, which are variations of Mumford’s example for [Formula: see text].