The analytic classification of plane curves with two branches (original) (raw)

2014, Mathematische Zeitschrift

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 Kähler differentials on the curve. * The first two authors were partially supported by CNPq grants The A-equivalence in B is induced by the action of the group A = Aut(C{t 1 })×Aut(C{t 2 })× Aut(C{X, Y }), as follows: (ρ 1 , ρ 2 , σ) • φ = [σ • φ 1 • ρ −1 1 , σ • φ 2 • ρ −1 2 ]. Our analysis will be splitted into two cases, namely, whether the two components of (f) have distinct tangents (the transversal case) or equal tangents. In what follows, we will denote by m i the multiplicity of f i , i = 1, 2. Case 1) Distinct tangents. In this case, by A-equivalence, we may assume that the tangent of the first component is (Y) and of the second one is (X), so that φ i = (x(t i), y(t i)), where ord t 1 x(t 1) < ord t 1 y(t 1) and ord t 2 x(t 2) > ord t 2 y(t 2). Case 2) Same tangent. In this case, by A-equivalence, we may assume that the common tangent