Jacobian curve of singular foliations

TL;DR

This paper investigates the topological properties of the jacobian curve of two foliations, providing a decomposition based on their similarity measured by Camacho-Sad indices, and unifying results on curve factorizations.

Contribution

It introduces a decomposition of the jacobian curve depending on foliation similarity, linking it to Camacho-Sad indices and unifying various factorization results.

Findings

Decomposition of the jacobian curve based on foliation similarity.

Relation of the jacobian curve to Camacho-Sad indices.

Unified approach to factorization of plane curves and polar curves.

Abstract

Topological properties of the jacobian curve ${\mathcal J}_{\mathcal{F},\mathcal{G}}$ of two foliations $\mathcal{F}$ and $\mathcal{G}$ are described in terms of invariants associated to the foliations. The main result gives a decomposition of the jacobian curve ${\mathcal J}_{\mathcal{F},\mathcal{G}}$ which depends on how similar are the foliations $\mathcal{F}$ and $\mathcal{G}$. The similarity between foliations is codified in terms of the Camacho-Sad indices of the foliations with the notion of collinear point or divisor. Our approach allows to recover the results concerning the factorization of the jacobian curve of two plane curves and of the polar curve of a curve or a foliation.