On characterizations of nondicritical generalized curve foliations

TL;DR

This paper characterizes nondicritical generalized curve foliations with fixed reduced separatrix, introducing Weierstrass forms and using toric resolution and GSV-index to provide new criteria and interpretations.

Contribution

It introduces Weierstrass forms for 1-forms and provides new characterizations of nondicritical generalized curve foliations with fixed reduced separatrix.

Findings

Characterization of nondicritical generalized curve foliations with fixed reduced separatrix.

Introduction of Weierstrass form for 1-forms.

Use of toric resolution and GSV-index for characterization.

Abstract

We characterize nondicrital generalized curve foliations with fixed reduced separatrix. Moreover, we give suficient conditions when a plane analytic curve is its reduced separatrix. For that, we introduce a distinguished expression for a given 1-form, called {\it Weierstrass form}. Then, using Weierstrass forms, we characterize the nondicritical generalized curve foliations: first, for foliations with monomial separatrix using toric resolution; second, for foliations with reduced separatrix, using the $GSV$-index. In this last case the characterization, which is our main result, could be interpreted in function of a polar of the foliation and a polar of its reduced separatrix.