The Hilbert scheme of space curves of small diameter

Kleppe, Jan Oddvar

Annales de l'Institut Fourier, Tome 56 (2006) no. 5, pp. 1297-1335.

Cet article concerne des courbes gauches $C$ de degré $d$ et de genre $g$, d’idéal homogène $I$ et de module de Rao $M={\mathrm{H}}_{*}^1\left(\tilde{I}\right)$ ; les résultats principaux portent sur les courbes qui vérifient ${}_0{\mathrm{Ext}}_R^2(M,M)=0$ (e.g. de diamètre, $\mathrm{diam}M\le 2$). Pour de telles courbes nous trouvons des conditions nécessaires et suffisantes pour être non obstruées et nous calculons la dimension du schéma de Hilbert, $\mathrm{H}(d,g)$ en $\left(C\right)$ 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 $\mathrm{H}(d,g)$ qui contiennent une courbe obstruée de diamètre 1. Nous prouvons aussi que chaque composante irréductible de $\mathrm{H}(d,g)$ est réduite au cas de diamètre 1.

This paper studies space curves $C$ of degree $d$ and arithmetic genus $g$, with homogeneous ideal $I$ and Rao module $M={\mathrm{H}}_{*}^1\left(\tilde{I}\right)$, whose main results deal with curves which satisfy ${}_0{\mathrm{Ext}}_R^2(M,M)=0$ (e.g. of diameter, $\mathrm{diam}M\le 2$). For such curves we find necessary and sufficient conditions for unobstructedness, and we compute the dimension of the Hilbert scheme, $\mathrm{H}(d,g)$, at $\left(C\right)$ 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 $\mathrm{H}(d,g)$ which contains an obstructed diameter one curve. We also show that every irreducible component of $\mathrm{H}(d,g)$ is reduced in the diameter one case.

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