[Obstructions à déformer des courbes sur une variété de dimension 3, II : Déformations des courbes dégénérées sur une variété de del Pezzo]
Nous étudions le schéma de Hilbert des courbes lisses connexes sur une variété de del Pezzo lisse de dimension 3. Nous montrons qu’aucune courbe dégénérée, c’est-à-dire, aucune courbe contenue dans une section hyperplane de , se déforme en une courbe non-dégénérée, si les deux conditions suivantes sont satisfaites : (i) et (ii) pour chaque droite sur telle que , le fibré normal de dans est trivial. Par conséquent, nous prouvons un analogue (pour ) d’une conjecture de J. O. Kleppe, qui concerne les composantes non-réduites du schéma de Hilbert des courbes dans l’espace projectif de dimension 3.
We study the Hilbert scheme of smooth connected curves on a smooth del Pezzo -fold . We prove that any degenerate curve , i.e. any curve contained in a smooth hyperplane section of , does not deform to a non-degenerate curve if the following two conditions are satisfied: (i) and (ii) for every line on such that , the normal bundle is trivial (i.e. ). As a consequence, we prove an analogue (for ) of a conjecture of J. O. Kleppe, which is concerned with non-reduced components of the Hilbert scheme of curves in the projective -space .
Keywords: Hilbert scheme, infinitesimal deformation, del Pezzo variety
Mot clés : schéma de Hilbert, déformations infinitésimales, variété de del Pezzo
Nasu, Hirokazu 1
@article{AIF_2010__60_4_1289_0,
author = {Nasu, Hirokazu},
title = {Obstructions to deforming curves on a $3$-fold, {II:} {Deformations} of degenerate curves on a del {Pezzo} $3$-fold},
journal = {Annales de l'Institut Fourier},
pages = {1289--1316},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {60},
number = {4},
year = {2010},
doi = {10.5802/aif.2555},
mrnumber = {2722242},
zbl = {1198.14004},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2555/}
}
TY - JOUR AU - Nasu, Hirokazu TI - Obstructions to deforming curves on a $3$-fold, II: Deformations of degenerate curves on a del Pezzo $3$-fold JO - Annales de l'Institut Fourier PY - 2010 SP - 1289 EP - 1316 VL - 60 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2555/ DO - 10.5802/aif.2555 LA - en ID - AIF_2010__60_4_1289_0 ER -
%0 Journal Article %A Nasu, Hirokazu %T Obstructions to deforming curves on a $3$-fold, II: Deformations of degenerate curves on a del Pezzo $3$-fold %J Annales de l'Institut Fourier %D 2010 %P 1289-1316 %V 60 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2555/ %R 10.5802/aif.2555 %G en %F AIF_2010__60_4_1289_0
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. https://aif.centre-mersenne.org/articles/10.5802/aif.2555/
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