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-open set of a projective space of dimension d ² + 4d + 2 on which Aut([Formula presented]) acts. We show that there are exactly two orbits, [Formula presented] and [Formula presented], of minimal dimension 6, necessarily closed in F(d). This generalizes known results in degrees 2 and 3. We deduce that an orbit [Formula presented] of an element [Formula presented] of dimension 7 is closed in F(d) if and only if [Formula presented] 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 [Formula presented] and unlike the situation in degree 2. On the other hand, we introduce the notion of the basin of attraction of a foliation [Formula presented] as the set of [Formula presented] such that [Formula presented]. We show that the basin of attraction [Formula presented], respectively [Formula presented], contains a quasi-projective subvariety of F(d) of dimension greater than or equal to dim F(d) — (d — 1), respectively dim F(d) — (d — 3). In particular, we obtain that the basin [Formula presented] contains a nonempty Zariski-open subset of F(3). This is an analog in degree 3 of a result on foliations of degree 2 due to Cerveau, Deserti, Garba Belko and Meziani.