Normal forms of foliations and curves defined by a function with a generic tangent cone

TL;DR

This paper characterizes the moduli spaces of foliations defined by analytic functions with multiple branches, establishes normal forms, and demonstrates that the associated distribution is rationally integrable, providing a complete set of invariants for generic plane curves.

Contribution

It describes the moduli spaces of such foliations, specifies normal forms, and proves the rational integrability of the distribution related to the equivalence of foliations by their invariant curves.

Findings

Normal forms for foliations with multiple branches are specified.

The distribution on the moduli space is proven to be rationally integrable.

A complete system of invariants extending cross-ratios is provided.

Abstract

We first describe the local and global moduli spaces of germs of foliations defined by analytic functions in two variables with p transverse smooth branches, and with integral multiplicities (in the univalued holomorphic case) or complex multiplicities (in the multivalued ''Darboux'' case). We specify normal forms in each class. Then we study on these moduli space the distribution C induced by the following equivalence relation: two points are equivalent if and only if the corresponding foliations have the same analytic invariant curves up to analytical conjugacy. Therefore, the space of leaves of C is the moduli space of curves. We prove that C is rationally integrable. These rational integrals give a complete system of invariants for these generic plane curves, which extend the well-known cross-ratios between branches.