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

TL;DR

This paper investigates how the Cremona group acts on holomorphic foliations of the complex projective plane, focusing on degree aberrations, numerical invariance, and their geometric implications, revealing complex behaviors and special structures.

Contribution

It introduces the concept of degree aberration in the Cremona action on foliations and explores the relationship between numerical invariance and transversal structures in low degrees.

Findings

Characterization of fixed points by Cantat-Favre

Identification of degree aberration phenomena

Link between numerical invariance and transversal structures

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.