The analytic classification of plane curves with two branches

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 Kähler differentials on the curve. * The first two authors were partially supported by CNPq grantsThe 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: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

Standard Bases for fractional ideals of the local ring of an algebroid curve

“…In fact, if we apply the algorithms presented in [13], we obtain that B ′ 1 = {x, y, z, h In this section, we will consider the module of Kähler differentials Ω O/K for an algebroid plane curve. In the analytical case, Ω O/C is an important example of fractional ideal since the relative ideal associated to it plays a central role in the analytic classification problem, as we can see in [14] for irreducible plane curves and in [15] for plane curves with two branches.…”

“…We remark that in [14] and [15] the authors considered Λ ∩ N r (r = 1, 2) as the main ingredient to proceed an answer to the analytic classification problem for curves with one and two branches. In addition, as we mentioned in Introduction, the set Λ is related to the values of the module of logarithmic residues along a complete intersection curve Q and to the set of values of the Jacobian ideal of Q.…”

“…We can consider Ω O/C as a fractional ideal of O and its set of values Λ allows us to stratify the topological class. This is an approach that has been used in the analytic classification problem (see [14] for the irreducible case and [15] for curves with two branches).…”

On the values of logarithmic residues along curves

“…Let Ω 1 C be the module of Kähler differentials along a reduced complete intersection curve C. The values of J C and the values of Ω 1 C satisfy: val(J C ) = γ + val(Ω 1 C ) − 1. The set of values of Kähler differentials is a major ingredient used in [HH11] and [HHH15] to study the problem of the analytic classification of plane curves with one or two branches.…”

Multiplicity, regularity and blow-spherical equivalence of complex analytic sets

“…More about classification of complex analytic surfaces can be found in [5], [12], [11], [10], [6] and [7]. We have also recent studies about classification of complex analytic curves, see for example [8], [31], [32] and [33].…”