Géométrie classique de certains feuilletages quadratiques

D. Cerveau, J. Déserti, D. Garba Belko, R. Meziani

Published 2009-02-05, updated 2009-10-14Version 6

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 $6.$