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

Abstract

The Cremona group of birational transformations of ${\mathbb{P}}_{\mathbb{C}}^2$ 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 $(\varphi ,\mathcal{F})\in \mathrm{Bir}({\mathbb{P}}_{\mathbb{C}}^2)\times \mathbb{F}(2)$ for which $\mathrm{deg}{\varphi }^{\ast }\mathcal{F}<(°\mathcal{F}+1)\mathrm{deg}\varphi +\mathrm{deg}\varphi -2$, the generic degree of such pull-back. We introduce the notion of numerical invariance ($\mathrm{deg}{\varphi }^{\ast }\mathcal{F}=°\mathcal{F}$) and relate it in small degrees to the existence of transversal structure for the considered foliations.

The proof of Proposition 5.3 has been simplified