Action of the Cremona group on foliations on $\mathbb{P}^2_\mathbb{C}$: some curious facts

Abstract: The Cremona group of birational transformations of $\mathbb{P}^2_\mathbb{C}$ acts on the space $\mathbb{F}(2)$ of holomorphic foliations on the complex projective plane. Since this action is not compatible with the natural graduation of $\mathbb{F}(2)$ by the degree, its description is complicated. The fixed points of the action are essentially described by Cantat-Favre in \cite{CF}. In that paper we are interested in problems of "aberration of the degree" that is pairs $(\phi,\mathcal{F})\in\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})\times\mathbb{F}(2)$ for which $\deg\phi^*\mathcal{F}<(\deg\mathcal{F}+1)\deg\phi+\deg\phi-2$, the generic degree of such pull-back. We introduce the notion of numerical invariance ($\deg\phi^*\mathcal{F}=\deg\mathcal{F}$) and relate it in small degrees to the existence of transversal structure for the considered foliations.