Obstructions to deforming curves on a $3$-fold, II: Deformations of degenerate curves on a del Pezzo $3$-fold

Nasu, Hirokazu

Annales de l'Institut Fourier, Tome 60 (2010) no. 4, pp. 1289-1316.

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

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

[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 $\mathrm{Hilb}{\mathbb{P}}^3$, 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 ${\mathbf{P}}^n$, 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