# Vector flows and the analytic moduli of singular plane branches

## TL;DR

This paper offers a geometric proof that simplifies the classification of singular plane branches by reducing their parametrizations, addressing a key gap in Zariski's approach to the moduli problem.

## Contribution

It provides a geometric elementary proof that plane branches can be simplified to short parametrizations, filling a crucial gap in Zariski's moduli problem solution.

## Key findings

- Established the existence of short parametrizations for plane branches
- Connected vector flows with the analytic moduli of singularities
- Filled a gap in Zariski's approach to the moduli problem

## Abstract

We provide a geometric elementary proof of the fact that an analytic plane branch is analytically equivalent to one whose terms corresponding to contacts with holomorphic one-forms -- except for Zariski's $\lambda$-invariant -- are zero (so called "short parametrizations"). This is the main step missed by Zariski in his attempt to solve the moduli problem.