The Hilbert Scheme of Buchsbaum space curves

Abstract

We consider the Hilbert scheme H(d,g) of space curves C with homogeneous ideal I(C):=H * 0 (ℐ C ) and Rao module M:=H * 1 (ℐ C ). By taking suitable generizations (deformations to a more general curve) C ′ of C, we simplify the minimal free resolution of I(C) by e.g making consecutive free summands (ghost-terms) disappear in a free resolution of I(C ′ ). Using this for Buchsbaum curves of diameter one (M v ≠0 for only one v), we establish a one-to-one correspondence between the set 𝒮 of irreducible components of H(d,g) that contain (C) and a set of minimal 5-tuples that specializes in an explicit manner to a 5-tuple of certain graded Betti numbers of C related to ghost-terms. Moreover we almost completely (resp. completely) determine the graded Betti numbers of all generizations of C (resp. all generic curves of 𝒮), and we give a specific description of the singular locus of the Hilbert scheme of curves of diameter at most one. We also prove some semi-continuity results for the graded Betti numbers of any space curve under some assumptions.