Obstructions to deforming curves on a 3-fold, II: Deformations of degenerate curves on a del Pezzo 3-fold
Annales de l'Institut Fourier, Volume 60 (2010) no. 4, p. 1289-1316

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 .

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.

DOI : https://doi.org/10.5802/aif.2555
Classification:  14C05,  14H10,  14D15
Keywords: Hilbert scheme, infinitesimal deformation, del Pezzo variety
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {4},
     year = {2010},
     pages = {1289-1316},
     doi = {10.5802/aif.2555},
     mrnumber = {2722242},
     zbl = {1198.14004},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 60 (2010) no. 4, pp. 1289-1316. doi : 10.5802/aif.2555. http://www.numdam.org/item/AIF_2010__60_4_1289_0/

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