Obstructions to deforming curves on a 3-fold, I: A generalization of Mumford's example and an application to Hom schemes

We give a sufficient condition for a first order infinitesimal deformation of a curve on a 3-fold to be obstructed. As application we construct generically non-reduced components of the Hilbert schemes of uniruled 3-folds and the Hom scheme from a general curve of genus five to a general cubic 3-fold.