Obstructions to deforming curves on a 3-fold, II: Deformations of degenerate curves on a del Pezzo 3-fold

We study the Hilbert scheme Hilb V of smooth connected curves on a smooth del Pezzo 3-fold V . We prove that every degenerate curve C, i.e. every curve contained in a smooth hyperplane section S of V , does not deform to a non-degenerate curve if the following two conditions are satisfied: (i) χ(V, IC(S)) ≥ 1 and (ii) for every line ` on S such that ` ∩ C = ∅, the normal bundle N`/V is trivial (i.e. N`/V ' OP1⊕2). As a consequence, we prove an analogue (for Hilb V ) of a conjecture of J.O. Kleppe which is concerned with non-reduced components of the Hilbert scheme Hilb P3 of curves in the 3-dimensional projective space P3.