Action of the Cremona group on foliations on ℙℂ 2: Some curious facts

Abstract

Abstract The Cremona group of birational transformations of ℙℂ 2 acts on the space 𝔽(2) of holomorphic foliations on the complex projective plane. Since this action is not compatible with the natural graduation of 𝔽(2) by the degree, its description is complicated. The fixed points of the action are essentially described by Cantat and Favre in [J. Reine Angew. Math. 561 (2003), 199–235]. In that article we are interested in problems of “aberration of the degree”, that is, pairs ( φ , ℱ ) $(\phi ,\mathcal {F})$ from Bir ( ℙ ℂ 2 ) × 𝔽 ( 2 ) ${\mathrm {Bir}(\mathbb {P}^2_\mathbb {C})\times \mathbb {F}(2)}$ for which deg φ * ℱ = deg ℱ < ( deg ℱ + 1 ) deg φ + deg φ - 2 $\deg \phi ^*\mathcal {F}=\deg \mathcal {F}<(\deg \mathcal {F}+1)\deg \phi +\deg \phi -2$ , the generic degree of such a pull-back. We introduce the notion of numerical invariance ( deg φ * ℱ = deg ℱ ${\deg \phi ^*\mathcal {F}=\deg \mathcal {F}}$ ) and relate it in small degrees to the existence of transversal structure for the considered foliations.