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  • Author: Dominique Cerveau x
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We study the classification of polynomial vector fields in two complex variables under the hypotheses that the singularities are isolated and the flow is complete. Normal forms are obtained for the case the generic orbit is diffeomorphic to ℂ. For the case the generic orbit is diffeomorphic to ℂ ∖ {0} and there is an affine singularity we classify the linear part of the vector field and prove the existence of entire linearization or first integral.


The Cremona group of birational transformations of ℙ 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,) from Bir (2)×𝔽(2) for which degφ*=deg<(deg+1)degφ+degφ-2, the generic degree of such a pull-back. We introduce the notion of numerical invariance (degφ*=deg) and relate it in small degrees to the existence of transversal structure for the considered foliations.