1 Se p 20 06 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 l on S such that l ∩ C = ∅, the normal bundle Nl/V is trivial (i.e. Nl/V ≃ OP1 ). 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.