Geometry of certain foliations on the complex projective plane

Let d≥2 be an integer. The set F(d) of foliations of degree d on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension d2+4d+2 on which Aut(P2C) acts. We show that there are exactly two orbits O(Fd1) and O(Fd2) of minimal dimension 6 , necessarily closed in F(d) . This generalizes known results in degrees 2 and 3. We deduce that an orbit O(F) of an element F∈F(d) of dimension 7 is closed in F(d) if and only if Fdi∉O(F)¯¯¯¯¯¯¯¯¯¯¯¯ for i=1,2. This allows us to show that in any degree d≥3 there are closed orbits in F(d) other than the orbits O(Fd1) and O(Fd2), unlike the situation in degree 2. On the other hand, we introduce the notion of the basin of attraction B(F) of a foliation F∈F(d) as the set of G∈F(d) such that F∈O(G)¯¯¯¯¯¯¯¯¯¯¯. We show that the basin of attraction B(Fd1) , resp. B(Fd2) , contains a quasi-projective subvariety of F(d) of dimension greater than or equal to dimF(d)−(d−1) , resp. dimF(d)−(d−3) . In particular, we obtain that the basin B(F32) contains a non-empty Zariski open subset of F(3) . This is an analog in degree 3 of a result on foliations of degree 2 due to Cerveau, Déserti, Garba Belko and Meziani.