Second type foliations of codimension one

Abstract

In this article, for holomorphic foliations of codimension one at $({\mathbb{C}}^3,0)$, we define the family of second type foliations. This is formed by foliations having, in the reduction process by blow-up maps, only well oriented singularities, meaning that the reduction divisor does not contain weak separatrices of saddle-node singularities. We prove that the reduction of singularities of a non-dicritical foliation of second type coincides with the desingularization of its set of separatrices.

15 pages, 1 figure