The Hilbert scheme of space curves of small diameter

Jan Oddvar Kleppe[1]

  • [1] Oslo University College Faculty of Engineering Pb. 4 St. Olavs plass 0130, Oslo (Norway)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 5, page 1297-1335
  • ISSN: 0373-0956

Abstract

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This paper studies space curves C of degree d and arithmetic genus g , with homogeneous ideal I and Rao module M = H * 1 ( I ˜ ) , whose main results deal with curves which satisfy 0 Ext R 2 ( M , M ) = 0 (e.g. of diameter, diam M ≤ 2 ). For such curves we find necessary and sufficient conditions for unobstructedness, and we compute the dimension of the Hilbert scheme, H ( d , g ) , at ( C ) under the sufficient conditions. In the diameter one 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 minimal resolution of I . More generally by taking suitable deformations of C we show how to kill repeated direct free factors (“ghost-terms”) in the minimal resolution of I , leading to a rather concrete description of the number of irreducible components of H ( d , g ) which contains an obstructed diameter one curve. We also show that every irreducible component of H ( d , g ) is reduced in the diameter one case.

How to cite

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Kleppe, Jan Oddvar. "The Hilbert scheme of space curves of small diameter." Annales de l’institut Fourier 56.5 (2006): 1297-1335. <http://eudml.org/doc/10178>.

@article{Kleppe2006,
abstract = {This paper studies space curves $C$ of degree $d$ and arithmetic genus $g$, with homogeneous ideal $I$ and Rao module $M = \textrm\{H\}_\{*\}^1(\tilde\{I\})$, whose main results deal with curves which satisfy $ \{_\{0\}\!\textrm\{Ext\}_R^2\}(M ,M ) = 0 $ (e.g. of diameter, $\textrm\{diam\} M \le 2$). For such curves we find necessary and sufficient conditions for unobstructedness, and we compute the dimension of the Hilbert scheme, $\textrm\{H\}(d,g)$, at $(C)$ under the sufficient conditions. In the diameter one 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 minimal resolution of $I$. More generally by taking suitable deformations of $C$ we show how to kill repeated direct free factors (“ghost-terms”) in the minimal resolution of $I$, leading to a rather concrete description of the number of irreducible components of $\textrm\{H\}(d,g)$ which contains an obstructed diameter one curve. We also show that every irreducible component of $\textrm\{H\}(d,g)$ is reduced in the diameter one case.},
affiliation = {Oslo University College Faculty of Engineering Pb. 4 St. Olavs plass 0130, Oslo (Norway)},
author = {Kleppe, Jan Oddvar},
journal = {Annales de l’institut Fourier},
keywords = {Hilbert scheme; space curve; Buchsbaum curve; unobstructedness; cup-product; graded Betti numbers; ghost term; linkage; normal module; postulation Hilbert scheme; ghost terms},
language = {eng},
number = {5},
pages = {1297-1335},
publisher = {Association des Annales de l’institut Fourier},
title = {The Hilbert scheme of space curves of small diameter},
url = {http://eudml.org/doc/10178},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Kleppe, Jan Oddvar
TI - The Hilbert scheme of space curves of small diameter
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 5
SP - 1297
EP - 1335
AB - This paper studies space curves $C$ of degree $d$ and arithmetic genus $g$, with homogeneous ideal $I$ and Rao module $M = \textrm{H}_{*}^1(\tilde{I})$, whose main results deal with curves which satisfy $ {_{0}\!\textrm{Ext}_R^2}(M ,M ) = 0 $ (e.g. of diameter, $\textrm{diam} M \le 2$). For such curves we find necessary and sufficient conditions for unobstructedness, and we compute the dimension of the Hilbert scheme, $\textrm{H}(d,g)$, at $(C)$ under the sufficient conditions. In the diameter one 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 minimal resolution of $I$. More generally by taking suitable deformations of $C$ we show how to kill repeated direct free factors (“ghost-terms”) in the minimal resolution of $I$, leading to a rather concrete description of the number of irreducible components of $\textrm{H}(d,g)$ which contains an obstructed diameter one curve. We also show that every irreducible component of $\textrm{H}(d,g)$ is reduced in the diameter one case.
LA - eng
KW - Hilbert scheme; space curve; Buchsbaum curve; unobstructedness; cup-product; graded Betti numbers; ghost term; linkage; normal module; postulation Hilbert scheme; ghost terms
UR - http://eudml.org/doc/10178
ER -

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