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hal-00459180, version 1

## Géométrie classique de certains feuilletages quadratiques

Dominique Cerveau 1, Julie Déserti 123, Djibrilla Garba Belko 4, Rafik Meziani 56

(2009-10-14)

Abstract: 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 $6.$

• 1:  Institut de Recherche Mathématique de Rennes (IRMAR)
• 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
• 2:  Institut de Mathématiques de Jussieu
• Université Paris VII - Paris Diderot
• 3:  Institut de Mathématiques de Jussieu
• Université Paris VII - Paris Diderot
• 4:  Faculté des Sciences
• Université Abdou Moumouni
• 5:  Département de Mathématiques
• Université Ibn Tofail
• 6:  Département de Mathématiques
• Université Ibn Tofail
• Domain : Mathematics/Dynamical Systems
Mathematics/Algebraic Geometry
• Comment : 26 pages – 14 figures – in french – for figures with higher resolution see http://people.math.jussieu.fr/~deserti/publications

• hal-00459180, version 1
• oai:hal.archives-ouvertes.fr:hal-00459180
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• Submitted on: Tuesday, 23 February 2010 14:07:29
• Updated on: Tuesday, 23 March 2010 15:11:53