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.

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: 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, Volume 60 (2010) no. 4, pp. 1289-1316. doi : 10.5802/aif.2555. https://aif.centre-mersenne.org/articles/10.5802/aif.2555/

[1] Curtin, D. Obstructions to deforming a space curve, Trans. Amer. Math. Soc., Volume 267 (1981), pp. 83-94 | DOI | MR | Zbl

[2] Ellia, Ph. D’autres composantes non réduites de Hilb 3 , Math. Ann., Volume 277 (1987), pp. 433-446 | DOI | MR | Zbl

[3] Fløystad, G. Determining obstructions for space curves, with applications to non-reduced components of the Hilbert scheme, J. Reine Angew. Math., Volume 439 (1993), pp. 11-44 | DOI | MR | Zbl

[4] Fujita, T. On the structure of polarized manifolds with total deficiency one. I, J. Math. Soc. Japan, Volume 32 (1980), pp. 709-725 | DOI | MR | Zbl

[5] Fujita, T. On the structure of polarized manifolds with total deficiency one. II, J. Math. Soc. Japan, Volume 33 (1981), pp. 415-434 | DOI | MR | Zbl

[6] Iskovskih, V. A. Fano threefolds. I, Math. USSR-Izvstija, Volume 11 (1977) no. 3, pp. 485-527 | DOI | MR | Zbl

[7] Iskovskih, V. A. Anticanonical models of three-dimensional algebraic varieties, Current problems in mathematics, J. Soviet Math., Volume 13 (1980), pp. 745-814 | DOI | MR | Zbl

[8] Kleppe, Jan O. Non-reduced components of the Hilbert scheme of smooth space curves., Space curves, Proc. Conf., Rocca di Papa/Italy 1985, Lect. Notes in Math. 1266, 181–207, 1987 | MR | Zbl

[9] Kleppe, Jan O. Liaison of families of subschemes in P n , Algebraic curves and projective geometry (Trento, 1988) (Lecture Notes in Math.), Volume 1389, Springer, Berlin, 1989, pp. 128-173 | MR | Zbl

[10] Kleppe, Jan O. The Hilbert scheme of space curves of small diameter, Annales de l’institut Fourier, Volume 56 (2006) no. 5, pp. 1297-1335 | DOI | Numdam | MR | Zbl

[11] Kollár, J. Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 32, Springer-Verlag, Berlin, 1996 | MR | Zbl

[12] Manin, Y. I. Cubic forms, North-Holland Mathematical Library, 4, North-Holland Publishing Co., Amsterdam, 1986 (Algebra, geometry, arithmetic, Translated from the Russian by M. Hazewinkel) | MR | Zbl

[13] Mukai, Shigeru; Nasu, Hirokazu Obstructions to deforming curves on a 3-fold. I. A generalization of Mumford’s example and an application to Hom schemes, J. Algebraic Geom., Volume 18 (2009) no. 4, pp. 691-709 | DOI | MR | Zbl

[14] Mumford, D. Further pathologies in algebraic geometry, Amer. J. Math., Volume 84 (1962), pp. 642-648 | DOI | MR | Zbl

[15] Nasu, Hirokazu Obstructions to deforming space curves and non-reduced components of the Hilbert scheme, Publ. Res. Inst. Math. Sci., Volume 42 (2006) no. 1, pp. 117-141 | DOI | MR | Zbl

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