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