The Hilbert scheme of space curves of small diameter
Annales de l'Institut Fourier, Volume 56 (2006) no. 5, pp. 1297-1335.

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, diamM2). 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.

Cet article concerne des courbes gauches C de degré d et de genre g, d’idéal homogène I et de module de Rao M=H * 1 (I ˜) ; les résultats principaux portent sur les courbes qui vérifient 0 Ext R 2 (M,M)=0 (e.g. de diamètre, diamM2). Pour de telles courbes nous trouvons des conditions nécessaires et suffisantes pour être non obstruées et nous calculons la dimension du schéma de Hilbert, 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 H(d,g) qui contiennent une courbe obstruée de diamètre 1. Nous prouvons aussi que chaque composante irréductible de H(d,g) est réduite au cas de diamètre 1.

DOI: 10.5802/aif.2214
Classification: 14C05,  14H50,  14B10,  14B15,  13D10,  13D02,  13D07,  13C40
Keywords: Hilbert scheme, space curve, Buchsbaum curve, unobstructedness, cup-product, graded Betti numbers, ghost term, linkage, normal module, postulation Hilbert scheme
Kleppe, Jan Oddvar 1

1 Oslo University College Faculty of Engineering Pb. 4 St. Olavs plass 0130, Oslo (Norway)
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Kleppe, Jan Oddvar. The Hilbert scheme of space curves of small diameter. Annales de l'Institut Fourier, Volume 56 (2006) no. 5, pp. 1297-1335. doi : 10.5802/aif.2214. http://archive.numdam.org/articles/10.5802/aif.2214/

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