Obstructions to deforming curves on a 3-fold, II: Deformations of degenerate curves on a del Pezzo 3-fold
[Obstructions à déformer des courbes sur une variété de dimension 3, II : Déformations des courbes dégénérées sur une variété de del Pezzo]
Annales de l'Institut Fourier, Tome 60 (2010) no. 4, pp. 1289-1316.

Nous étudions le schéma de Hilbert Hilb sc V des courbes lisses connexes sur une variété de del Pezzo lisse V de dimension 3. Nous montrons qu’aucune courbe C dégénérée, c’est-à-dire, aucune courbe C contenue dans une section hyperplane S de V, se déforme en une courbe non-dégénérée, si les deux conditions suivantes sont satisfaites  : (i) χ(V, C (S))1 et (ii) pour chaque droite sur S telle que C=, le fibré normal N /V de dans V est trivial. Par conséquent, nous prouvons un analogue (pour Hilb sc V) d’une conjecture de J. O. Kleppe, qui concerne les composantes non-réduites du schéma de Hilbert Hilb sc 3 des courbes dans l’espace projectif 3 de dimension 3.

We study the Hilbert scheme Hilb sc V of smooth connected curves on a smooth del Pezzo 3-fold V. We prove that any degenerate curve C, i.e. any curve C contained in a smooth hyperplane section S of V, does not deform to a non-degenerate curve if the following two conditions are satisfied: (i) χ(V, C (S))1 and (ii) for every line on S such that C=, the normal bundle N /V is trivial (i.e.  N /V 𝒪 1 2 ). As a consequence, we prove an analogue (for Hilb sc V) of a conjecture of J. O. Kleppe, which is concerned with non-reduced components of the Hilbert scheme Hilb sc 3 of curves in the projective 3-space 3 .

DOI : 10.5802/aif.2555
Classification : 14C05, 14H10, 14D15
Keywords: Hilbert scheme, infinitesimal deformation, del Pezzo variety
Mot clés : schéma de Hilbert, déformations infinitésimales, variété de del Pezzo
Nasu, Hirokazu 1

1 Kyoto University Research Institute for Mathematical Sciences Kyoto 606-8502 (Japan)
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Nasu, Hirokazu. Obstructions to deforming curves  on a $3$-fold, II: Deformations of degenerate curves on a del Pezzo $3$-fold. Annales de l'Institut Fourier, Tome 60 (2010) no. 4, pp. 1289-1316. doi : 10.5802/aif.2555. http://www.numdam.org/articles/10.5802/aif.2555/

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