Jan O. Kleppe
Mathematics Subject Classification (1991): 14C05, 14H50, 14B10, 14B15, 13D02
In this paper we study space curves C (with saturated homogeneous ideal I ) of degree d and arithmetic genus g whose diameter is two or less (or more generally, one may suppose that Ext2(M, M), M = H*1(I ) the Hartshorne-Rao module, vanishes in degree zero). For such curves C we find necessary and sufficient conditions for unobstructedness (i.e. the Hilbert scheme H(d,g) of space curves is smooth at C ), in which case we also compute the dimension of H(d,g) at C. In the diameter 1 case, the necessary and sufficient conditions coincide, and the unobstructedness of C turns out to be equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of I. For some classes of obstructed curves C, we partially compute the equations of the singularity of H(d,g) at C.
Moreover for any curve C we show how to kill certain repeated direct factors in the minimal resolution of I ("ghost-terms") by taking a suitable deformation. For Buchsbaum curves of diameter at most 2, we simplify in this way the minimal resolution further. It follows that the graded Betti numbers mentioned above of a generic curve vanish. Any irreducible component of H(d,g) is therefore reduced (generically smooth) in the diameter 1 case.
In an appendix we study maximal families W of space curves on a smooth cubic surface of any diameter with the restriction M1 = 0 and M3 non-zero, and we prove, under weak further conditions, that the closure of W in H(d,g) is a non-reduced component provided g > 3d - 19, thus proving a certain conjecture in this direction in most cases.