Folheações efetivas no plano complexo

Abstract

Many properties of a holomorphic foliation can be deduced from its set of separatrices. The Newton polygon is a geometric tool that encodes local information about analytic curves and holomorphic foliations. Several works explore the relationship between the Newton polygons of certain non-dicritical foliations and those of their separatrices. In this thesis, we propose to study this relationship in the context of dicritical foliations. To this end, we restrict our attention to a particular class of foliations, called effective foliations, characterized by having, in the exceptional divisor of their reduction of singularities, dicritical components of valence at most two. Foliations in this class admit a balanced equations of separatrices without poles. We show that, for an effective second type foliation, its Newton polygon coincides with the Newton polygon of its balanced equation of separatrices, thus generalizing results previously obtained for non-dicritical foliations in [14] and [29]. This approach has applications in the study of polar curves associated with a foliation. We present a generalization, for effective second type dicritical foliations, of the Polar Decomposition Theorem, proved in [10] for nondicritical generalized curve foliations. This result describes, from a topological perspective, the decomposition of polar curves into packages of branches, based on equisingularity data of the foliation’s separatrices.