Gacute;ométrie classique de certains feuilletages de degré deux

The set ℱ(2; 2) of foliations of degree 2 on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension 14 on which acts Aut(ℙ2(ℂ)). We classify, up to automorphisms of ℙ2(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 Aut(ℙ2(ℂ)) on 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 ℱ(2; 2) contains at least seven orbits of dimension 7 and exactly one orbit of dimension 6. © 2010 Springer.