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

Hirokazu Nasu[1]

  • [1] Kyoto University Research Institute for Mathematical Sciences Kyoto 606-8502 (Japan)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 4, page 1289-1316
  • ISSN: 0373-0956

Abstract

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We study the Hilbert scheme Hilb s c 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 , ℐ C ( S ) ) ≥ 1 and (ii) for every line ℓ on S such that ℓ ∩ C = ∅ , the normal bundle N ℓ / V is trivial (i.e.  N ℓ / V ≃ 𝒪 ℙ 1 ⊕ 2 ). As a consequence, we prove an analogue (for Hilb s c V ) of a conjecture of J. O. Kleppe, which is concerned with non-reduced components of the Hilbert scheme Hilb s c ℙ 3 of curves in the projective 3 -space ℙ 3 .

How to cite

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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 60.4 (2010): 1289-1316. <http://eudml.org/doc/116304>.

@article{Nasu2010,
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 $\mathbb\{P\}^3$.},
affiliation = {Kyoto University Research Institute for Mathematical Sciences Kyoto 606-8502 (Japan)},
author = {Nasu, Hirokazu},
journal = {Annales de l’institut Fourier},
keywords = {Hilbert scheme; infinitesimal deformation; del Pezzo variety},
language = {eng},
number = {4},
pages = {1289-1316},
publisher = {Association des Annales de l’institut Fourier},
title = {Obstructions to deforming curves on a $3$-fold, II: Deformations of degenerate curves on a del Pezzo $3$-fold},
url = {http://eudml.org/doc/116304},
volume = {60},
year = {2010},
}

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
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 4
SP - 1289
EP - 1316
AB - 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 $\mathbb{P}^3$.
LA - eng
KW - Hilbert scheme; infinitesimal deformation; del Pezzo variety
UR - http://eudml.org/doc/116304
ER -

References

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  10. Jan O. Kleppe, The Hilbert scheme of space curves of small diameter, Annales de l’institut Fourier 56 (2006), 1297-1335 Zbl1117.14006MR2273858
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  12. Y. I. Manin, Cubic forms, 4 (1986), North-Holland Publishing Co., Amsterdam Zbl0582.14010MR833513
  13. Shigeru Mukai, Hirokazu Nasu, Obstructions to deforming curves on a 3-fold. I. A generalization of Mumford’s example and an application to Hom schemes, J. Algebraic Geom. 18 (2009), 691-709 Zbl1181.14031MR2524595
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  15. Hirokazu Nasu, Obstructions to deforming space curves and non-reduced components of the Hilbert scheme, Publ. Res. Inst. Math. Sci. 42 (2006), 117-141 Zbl1100.14002MR2215438

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