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]

Nasu, Hirokazu

Annales de l'Institut Fourier, Tome 60 (2010), p. 1289-1316 / Harvested from Numdam

Résumé

Nous étudions le schéma de Hilbert $\textrm{Hilb}^{sc} V$ 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) $\chi (V,\mathcal{I}_C(S))\ge 1$ et (ii) pour chaque droite $\ell $ sur $S$ telle que $\ell \cap C = \emptyset$, le fibré normal $N_{\ell /V}$ de $\ell $ dans $V$ est trivial. Par conséquent, nous prouvons un analogue (pour $\textrm{Hilb}^{sc} V$) d’une conjecture de J. O. Kleppe, qui concerne les composantes non-réduites du schéma de Hilbert $\textrm{Hilb}^{sc} \mathbb{P}^3$ des courbes dans l’espace projectif $\mathbb{P}^3$ de dimension 3.

Abstract

We study the Hilbert scheme $\textrm{Hilb}^{sc} V$ 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) $\chi (V,\mathcal{I}_C(S))\ge 1$ and (ii) for every line $\ell $ on $S$ such that $\ell \cap C = \emptyset$, the normal bundle $N_{\ell /V}$ is trivial (i.e.  $N_{\ell /V} \simeq {\mathcal{O}_{\mathbb{P}^1}}^{\oplus 2}$). As a consequence, we prove an analogue (for $\textrm{Hilb}^{sc} V$) of a conjecture of J. O. Kleppe, which is concerned with non-reduced components of the Hilbert scheme $\textrm{Hilb}^{sc} \mathbb{P}^3$ of curves in the projective $3$-space $\mathbb{P}^3$.

Détail

Publié le : 2010-01-01

MR 2722242 | Zbl 1198.14004

DOI : https://doi.org/10.5802/aif.2555
Classification:  14C05,  14H10,  14D15
Mots clés: schéma de Hilbert, déformations infinitésimales, variété de del Pezzo

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