Jacobian curve of singular foliations

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.