The Hilbert scheme of space curves of small diameter

[Le schéma de Hilbert des courbes de petit diamètre]

Kleppe, Jan Oddvar

Annales de l'Institut Fourier, Tome 56 (2006), p. 1297-1335 / Harvested from Numdam

Résumé

Cet article concerne des courbes gauches $C$ de degré $d$ et de genre $g$, d’idéal homogène $I$ et de module de Rao $M = \textrm{H}_{*}^1(\tilde{I})$ ; les résultats principaux portent sur les courbes qui vérifient $ {_{0}\!\textrm{Ext}_R^2}(M ,M ) = 0 $ (e.g. de diamètre, $\textrm{diam} M \le 2$). Pour de telles courbes nous trouvons des conditions nécessaires et suffisantes pour être non obstruées et nous calculons la dimension du schéma de Hilbert, $\textrm{H}(d,g)$ en $(C)$ sous des conditions suffisantes. Dans le cas du diamètre 1, les conditions nécessaires et suffisantes coïncident et la condition d’être non obstruée s’avère être équivalente à l’annulation de certains nombres de Betti gradués de la résolution libre minimale de $I$. Plus généralement, en prenant des déformations convenables de $C$, nous montrons comment éliminer les facteurs directs libres répétés (“termes-fantômes”) dans la résolution minimale de $I$, conduisant à une description relativement concrète du nombre des composantes irréductibles de $\textrm{H}(d,g)$ qui contiennent une courbe obstruée de diamètre 1. Nous prouvons aussi que chaque composante irréductible de $\textrm{H}(d,g)$ est réduite au cas de diamètre 1.

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.

Détail

Publié le : 2006-01-01

MR 2273858 | Zbl 1117.14006 | 2 citations dans Numdam

DOI : https://doi.org/10.5802/aif.2214
Classification:  14C05,  14H50,  14B10,  14B15,  13D10,  13D02,  13D07,  13C40
Mots clés: schéma de Hilbert, courbe de l’espace, courbe de Buschsbaum, non obstruction, nombres de Betti gradués, termes-fantômes, lien, module normal, schéma de Hilbert

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