Vector flows and the analytic moduli of singular plane branches

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.

Much improved proof with very detailed sections on contacts, Taylor expansions and n-th roots