The Analytic Classification of Irreducible Plane Curve Singularities

In 1973, Oscar Zariski gave a course at the École Polytechnique (cf. the lecture notes [23] or the monograph [24]), where he discussed in the local case the analogous problem to the construction of the moduli space of algebraic curves of genus g; that is, he proposed to search for moduli spaces with respect to analytic equivalence for germs of irreducible complex analytic plane curves having the same topological type. Zariski recognized that this problem was at that stage very difficult and concentrated his efforts on explicit calculations in some particular cases. Since then, the subject has substantially advanced and our purpose here is to further detail the solution of the moduli problem for plane branches given in [12], using the same framework as Zariski’s, adding to his methods singularity tools that were starting to blossom at this time. The results we present improve several ones found in the literature and shed light over many questions asked by Zariski, using techniques that may be useful to solve other relevant related questions. The exposition is kept as elementary as possible to highlight the beauty and simplicity of the solution of Zariski’s problem and since it is not intended to be a compendium on the subject, several results from algebra and singularity theory will be invoked, giving precise statements and references, where they may be found, without concern about quoting primary sources.