# Generic 1-parameter pertubations of a vector field with a singular point of codimension k

## TL;DR

This paper classifies generic one-parameter families of complex vector fields near a singular point of multiplicity k+1, providing normal forms, moduli space descriptions, and bifurcation diagrams.

## Contribution

It introduces a comprehensive classification of germs of generic 1-parameter vector field families with a singular point of codimension k, including normal forms and bifurcation analysis.

## Key findings

- Complete bifurcation diagram of ż = z^{k+1} - ε over CP1.
- Description of the modulus space for the unfolding.
- Almost unique normal forms for the vector fields.

## Abstract

We describe the equivalence classes of germs of generic 1-parameter families of complex vector fields z dot = omega_epsilon(z) on C unfolding a singular point of multiplicity k+1: omega_0 = z^{k+1} + o(z^{k+1}). The equivalence is under conjugacy by holomorphic change of coordinate and parameter. We provide a description of the modulus space and (almost) unique normal forms. As a preparatory step, we present the complete bifurcation diagram of the family of vector fields z dot = z^{k+1} - epsilon, over CP1.