The Hilbert Scheme of Buchsbaum space curves
Annales de l'Institut Fourier, Volume 62 (2012) no. 6, p. 2099-2130
We consider the Hilbert scheme H(d,g) of space curves C with homogeneous ideal I(C):=H * 0 ( C ) and Rao module M:=H * 1 ( 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.
Nous considérons le schéma de Hilbert H(d,g) des courbes C dont l’idéal homogène est I(C):=H * 0 ( C ) et le module de Rao M:=H * 1 ( C ). En prenant des générisations (déformations) convenables C de C on simplifie la résolution minimale libre de I(C). Par exemple, certains facteurs libres consécutifs vont disparaître dans une résolution libre de I(C ). En appliquant ceci à des courbes de Buchsbaum de diamètre 1 (M v 0 seulement pour une valeur de v), nous donnons une correspondance biunivoque entre l’ensemble 𝒮 des composantes irréductibles de H(d,g) qui contiennent (C) et un ensemble des quintuplets minimaux, qui se spécialise à un quintuple de nombres de Betti gradués de C. De plus nous déterminons presque complétement les nombres de Betti gradués de toutes les générisations de C, et nous donnons une description du lieu singulier du schéma de Hilbert des courbes de diamètre au plus égal à 1. Nous démontrons aussi des résultats de sémi-continuité pour les nombres de Betti gradués des courbes.
DOI : https://doi.org/10.5802/aif.2744
Classification:  14C05,  14H50,  14M06,  13D02,  13C40
Keywords: Hilbert scheme, space curve, Buchsbaum curve, graded Betti numbers, ghost term, linkage.
@article{AIF_2012__62_6_2099_0,
     author = {Kleppe, Jan O.},
     title = {The Hilbert Scheme of Buchsbaum space curves},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {6},
     year = {2012},
     pages = {2099-2130},
     doi = {10.5802/aif.2744},
     zbl = {1271.14007},
     mrnumber = {3060753},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2012__62_6_2099_0}
}
The Hilbert Scheme of Buchsbaum space curves. Annales de l'Institut Fourier, Volume 62 (2012) no. 6, pp. 2099-2130. doi : 10.5802/aif.2744. https://aif.centre-mersenne.org/item/AIF_2012__62_6_2099_0/

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