Infinitesimal adjunction and polar curves

TL;DR

This paper studies the equisingularity types of polar curves associated with foliations having a curve of separatrices, identifying specific singularities that determine the polar curve's type for general foliations using adjunction properties.

Contribution

It introduces the concept of kind singularities of curves and shows these types determine the polar curve's equisingularity for Zariski-general foliations, extending classical polar curve theory.

Findings

Identifies kind singularities that determine polar curve types

Uses adjunction properties to analyze polar curves

Shows the importance of foliation framework for generality

Abstract

The polar curves of foliations $\mathcal F$ having a curve $C$ of separatrices generalize the classical polar curves associated to hamiltonian foliations of $C$. As in the classical theory, the equisingularity type ${\wp}({\mathcal F})$ of a generic polar curve depends on the analytical type of ${\mathcal F}$, and hence of $C$. In this paper we find the equisingularity types $\epsilon (C)$ of $C$, that we call kind singularities, such that ${\wp}({\mathcal F})$ is completely determined by $\epsilon (C)$ for Zariski-general foliations $\mathcal F$. Our proofs are mainly based on the adjunction properties of the polar curves. The foliation-like framework is necessary, otherwise we do not get the right concept of general foliation in Zariski sense and, as we show by examples, the hamiltonian case can be out of the set of general foliations.