The Analytic Classification of Plane Curves with Two Branches

TL;DR

This paper introduces a new analytic invariant to classify plane curve singularities with two branches, providing their normal forms and advancing understanding of their equivalence classes.

Contribution

It presents a novel invariant linking tangent vectors and Kähler differentials, enabling the analytic classification of two-branch plane curve singularities.

Findings

Established normal forms for two-branch plane curve singularities

Introduced a new invariant relating tangent vectors to Kähler differentials

Enhanced the understanding of analytic equivalence classes

Abstract

In this paper we solve the problem of analytic classification of plane curves singularities with two branches by presenting their normal forms. This is accomplished by means of a new analytic invariant that relates vectors in the tangent space to the orbits under analytic equivalence in a given equisingularity class to K\"ahler differentials on the curve.