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

Abstract

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) the Euler characteristic of the twisted ideal sheaf I_C(S) of C is greater than or equal to one, and (ii) for every line E on S which is disjoint to C, the normal bundle of E in V is trivial. 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 P^3) of curves in the 3-dimensional projective space P^3.