On codimension one foliations with prescribed cuspidal separatrix

TL;DR

This paper develops a pre-normal form for certain codimension one holomorphic foliations with cuspidal separatrices and provides conditions for these foliations to be of generalized surface type in three dimensions.

Contribution

It introduces a pre-normal form for germs of foliations with cuspidal separatrices and establishes a sufficient condition for their classification as generalized surface type in three dimensions.

Findings

Constructed a pre-normal form for cuspidal separatrix foliations.

Provided a sufficient condition for generalized surface type classification.

Focused on quasi-homogeneous, three-dimensional cases.

Abstract

In this paper, we will construct a pre-normal form for germs of codimension one holomorphic foliation having a particular type of separatrix, of cuspidal type. We will also give a sufficient condition, in the quasi-homogeneous, three-dimensional case, for these foliations to be of generalized surface type.