Let $d\geq2$ be an integer. The set $\mathbf{ F}(d)$ 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 $d^2+4d+2$ on which $\mathrm{ Aut}(\mathbb P^2_{ \mathbb C})$ acts. We show that there are exactly two orbits $\mathcal{ O}(\mathcal{ F}_{ 1}^{ d})$ and $\mathcal{ O}(\mathcal{ F}_{ 2}^{ d})$ of minimal dimension $6$, necessarily closed in $\mathbf{ F}(d)$. This generalizes known results in degrees $2$ and $3.$ We deduce that an orbit $\mathcal{ O}(\mathcal{ F})$ of an element $\mathcal{ F}\in\mathbf{ F}(d)$ of dimension $7$ is closed in $\mathbf{ F}(d)$ if and only if $\mathcal{ F}_{ i}^{ d}\not\in\overline{ \mathcal{ O}(\mathcal{ F})}$ for $i=1,2.$ This allows us to show that in any degree $d\geq3$ there are closed orbits in $\mathbf F(d)$ other than the orbits $\mathcal{ O}(\mathcal{ F}_{ 1}^{ d})$ and $\mathcal{ O}(\mathcal{ F}_{ 2}^{ d}),$ unlike the situation in degree $2.$ On the other hand, we introduce the notion of the basin of attraction $\mathbf{ B}(\mathcal{ F})$ of a foliation $\mathcal{ F}\in\mathbf{ F}(d)$ as the set of $\mathcal{ G}\in\mathbf{ F}(d)$ such that $\mathcal{ F}\in\overline{ \mathcal{ O}(\mathcal{ G})}.$ We show that the basin of attraction $\mathbf{ B}(\mathcal{ F}_{ 1}^{ d})$, resp. $\mathbf{ B}(\mathcal{ F}_{ 2}^{ d})$, contains a quasi-projective subvariety of $\mathbf{ F}(d)$ of dimension greater than or equal to $\dim\mathbf{ F}(d)-(d-1)$, resp. $\dim \mathbf{ F}(d)-(d-3)$. In particular, we obtain that the basin $\mathbf{ B}(\mathcal{ F}_{ 2}^{ 3})$ contains a non-empty Zariski open subset of $\mathbf{ F}(3)$. This is an analog in degree $3$ of a result on foliations of degree $2$ due to Cerveau, D\'eserti, Garba Belko and Meziani.
Categories: math.DS math.CV math.DG