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| HAL: hal-00459180, version 1 |
| arXiv: 0902.0877 |
| Detailed view |  Export this paper |
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| Géométrie classique de certains feuilletages quadratiques |
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| Dominique Cerveau 1Julie Déserti 1, 2, 3 |
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| (2009-10-14) |
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| The set $\mathscr{F}(2;2)$ of quadratic foliations on the complex projective plane can be identified with a \textsc{Zariski}'s open set of a projective space of dimension 14 on which acts $\mathrm{Aut}(\mathbb{P}^2(\mathbb{C})).$ We classify, up to automorphisms of $\mathbb{P}^2(\mathbb{C}),$ quadratic foliations with only one singularity. There are only four such foliations up to conjugacy; whereas three of them have a dynamic which can be easily described the dynamic of the fourth is still mysterious. This classification also allows us to describe the action of $\mathrm{Aut}(\mathbb{P}^2(\mathbb{C}))$ on $\mathscr{F}(2;2).$ On the one hand we show that the dimension of the orbits is more than 6 and that there are exactly two orbits of dimension $6;$ on the other hand we obtain that the closure of the generic orbit in $\mathscr{F} (2;2)$ contains at least seven orbits of dimension~7 and exactly one orbit of dimension $6.$ |
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| 1: | Institut de Recherche Mathématique de Rennes (IRMAR) |
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CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne |
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| 2: | Institut de Mathématiques de Jussieu |
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Université Paris VII - Paris Diderot |
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| 3: | Institut de Mathématiques de Jussieu |
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Université Paris VII - Paris Diderot |
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| 4: | Faculté des Sciences |
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Université Abdou Moumouni |
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| 5: | Département de Mathématiques |
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Université Ibn Tofail |
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| 6: | Département de Mathématiques |
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Université Ibn Tofail |
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| Subject | : | Mathematics/Dynamical Systems Mathematics/Algebraic Geometry |
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| Fulltext link: |
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| http://fr.arXiv.org/abs/0902.0877 |
| hal-00459180, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00459180 | |
| oai:hal.archives-ouvertes.fr:hal-00459180 | |
| From: Marie-Annick Guillemer | |
| Submitted on: Tuesday, 23 February 2010 14:07:29 | |
| Updated on: Tuesday, 23 March 2010 15:11:53 | |
