The Hilbert Scheme of Buchsbaum space curves

Jan O. Kleppe[1]

  • [1] Oslo and Akershus University College of Applied Sciences Faculty of Technology, Art and Design Pb. 4, St. Olavs plass N-0130 Oslo, Norway

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 6, page 2099-2130
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Kleppe, Jan O.. "The Hilbert Scheme of Buchsbaum space curves." Annales de l’institut Fourier 62.6 (2012): 2099-2130. <http://eudml.org/doc/251108>.

@article{Kleppe2012,
abstract = {We consider the Hilbert scheme $\mathop \{H\}(d,g)$ of space curves $C$ with homogeneous ideal $I(C):=H_\{*\}^0(\mathcal\{I\}_\{C\})$ and Rao module $M:=H_\{*\}^1(\mathcal\{I\}_\{C\})$. By taking suitable generizations (deformations to a more general curve) $C^\{\prime\}$ 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 $I(C^\{\prime\})$. 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 $\mathcal\{S\}$ of irreducible components of $\mathop \{H\}(d,g)$ 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 $\mathcal\{S\}$), 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.},
affiliation = {Oslo and Akershus University College of Applied Sciences Faculty of Technology, Art and Design Pb. 4, St. Olavs plass N-0130 Oslo, Norway},
author = {Kleppe, Jan O.},
journal = {Annales de l’institut Fourier},
keywords = {Hilbert scheme; space curve; Buchsbaum curve; graded Betti numbers; ghost term; linkage},
language = {eng},
number = {6},
pages = {2099-2130},
publisher = {Association des Annales de l’institut Fourier},
title = {The Hilbert Scheme of Buchsbaum space curves},
url = {http://eudml.org/doc/251108},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Kleppe, Jan O.
TI - The Hilbert Scheme of Buchsbaum space curves
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 6
SP - 2099
EP - 2130
AB - We consider the Hilbert scheme $\mathop {H}(d,g)$ of space curves $C$ with homogeneous ideal $I(C):=H_{*}^0(\mathcal{I}_{C})$ and Rao module $M:=H_{*}^1(\mathcal{I}_{C})$. By taking suitable generizations (deformations to a more general curve) $C^{\prime}$ 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 $I(C^{\prime})$. 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 $\mathcal{S}$ of irreducible components of $\mathop {H}(d,g)$ 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 $\mathcal{S}$), 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.
LA - eng
KW - Hilbert scheme; space curve; Buchsbaum curve; graded Betti numbers; ghost term; linkage
UR - http://eudml.org/doc/251108
ER -

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