Classification and moduli spaces of dicritical singularities

TL;DR

This paper classifies certain holomorphic foliation singularities in complex surfaces, providing invariants, normal forms, and explicit examples of unfoldings, advancing the understanding of their moduli spaces and conjugacy classes.

Contribution

It introduces a complete set of analytic invariants for dicritical singularities with a single blow-up, and describes their moduli spaces and normal forms.

Findings

Complete analytic invariants for the singularities.

Explicit geometric realization of invariants.

Construction of universal unfoldings not obtainable by unfolding functions.

Abstract

In this paper we give complete analytic invariants for germs of holomorphic foliations in $(\mathbb{C}^2,0)$ that become regular after a single blow-up. Some of them describe the holonomy pseudogroup of the germ and are called transverse invariants. The other invariants lie in finite dimensional complex vector space. Such singularities admit separatrices tangent to any direction at the origin. When enough separatrices coincide with their tangent directions (a condition that can always be attained if the mutiplicity of the germ at the origin is at most four) we are able to describe and realize all the analytical invariants geometrically and provide analytic normal forms. As a consequence we prove that any two such germs sharing the same transverse invariants are conjugated by a very particular type of birational transformations. We also provide the first explicit examples of universal equisingular unfoldings of foliations that cannot be produced by unfolding functions. With these at hand we are able to explicitely parametrize families of analytically distinct foliations that share the same transverse invariants.