Folheações em P2 admitindo feixe linear polar redutível

Abstract

Given a foliation F on P2 , by fixing a line L P2 C, we define the polar pencil of F with axis L as the set of all polar curves of F with respect to points l 2 L. We begin this thesis by studying foliations F which admit a polar pencil whose generic element is reducible. To such an F we associate a primitive model, which is a foliation e F whose polar pencil, besides having irreducible generic element, is such that its curves are contained in those of the polar pencil of F. We establish geometric properties that relate a foliation F and its primitive model e F, such as relations between the Milnor numbers of their singularities. In the finalpart, we explore the concept of linear equivalence, two foliations being linearly equivalent if, and only if, they have the same polar pencil. By using this concept, we construct a twoparameter family of foliations on P2C with the same singular set and the same Milnor numbers at their singular points, and this shows that the singular set alone does not define a foliation. We also prove that the polar net defines in a unique way a foliation F in P2 C and, as a direct consequence, a foliation on P2C is uniquely defined by its singular subesqueme. Finally, we study common invariant varieties to two linearly equivalent foliations and explore its consequences.