Second type foliations of codimension one

TL;DR

This paper introduces the concept of second type foliations in codimension one holomorphic foliations at the origin in three-dimensional complex space, showing their reduction process aligns with desingularization of separatrices.

Contribution

It defines second type foliations and proves their singularity reduction process matches the desingularization of separatrices, advancing understanding of foliation singularities.

Findings

Second type foliations have only well oriented singularities during reduction.

Reduction of second type foliations coincides with desingularization of separatrices.

Provides a new framework for analyzing singularities in holomorphic foliations.

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.