The Hilbert Scheme of Buchsbaum space curves

TL;DR

This paper studies the structure of the Hilbert scheme of space curves, especially Buchsbaum curves of diameter one, by analyzing their minimal free resolutions, Betti numbers, and the geometry of their irreducible components.

Contribution

It establishes a correspondence between components of the Hilbert scheme and minimal 5-tuples of graded Betti numbers for Buchsbaum curves of diameter one, and characterizes their generic properties.

Findings

One-to-one correspondence between irreducible components and minimal 5-tuples.

Complete determination of Betti numbers for generic curves in each component.

Description of the singular locus of the Hilbert scheme for diameter at most one.

Abstract

We consider the Hilbert scheme H(d,g) of space curves C with homogeneous ideal I(C):=H_{*}^0(\sI_C) and Rao module M:=H_{*}^1(\sI_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 \ne 0 for only one v), we establish a one-to-one correspondence between the set \sS 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 \sS), 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.