Skip to main content
SpringerLink
Log in

The analytic classification of plane curves with two branches

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In this paper we solve effectively the problem of local analytic classification of plane curves singularities with two branches. We present normal forms for these singularities and show how to reduce them to their normal forms. This is accomplished by introducing 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bruce, J.W., Kirk, N.P., du Plessis, A.A.: Complete transversals and the classification of singularities. Nonlinearity 10(1), 253–275 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Campillo, A., Delgado, F., Gusein-Zade, S.M.: The extended semigroup of a plane curve singularity. Proc. Steklov Inst. (in Honour of V.I. Arnold) 221, 149–167 (1998)

  3. Delgado, F.: The semigroup of values of a curve singularity with several branches. Manusc. Math. 59, 347–374 (1987)

    Article  MATH  Google Scholar 

  4. Ebey, S.: The classification of singular points of algebraic curves. Trans. Am. Math. Soc. 118, 454–471 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  5. Garcia, A.: Semigroups associated to singular points of plane curves. J. Reine Angew. Math. 336, 165–184 (1982)

    MATH  MathSciNet  Google Scholar 

  6. Gibson, C.G.: Singular Points of Smooth Mappings. Research Notes in Mathematics, vol. 25. Pitman, London (1979)

  7. Genzmer, Y., Paul, E.: Normal forms of foliations and curves defined by a function with a generic tangent cone. Mosc. Math. J. 11(1), 41–72 (2011)

    MATH  MathSciNet  Google Scholar 

  8. Hefez, A., Hernandes, M.E.: The analytic classification of plane branches. Bull. Lond. Math. Soc. 43(2), 289–298 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kolgushkin, P.A., Sadykov, R.R.: Simple singularities of multigerms of curves. Rev. Mat. Complut. XIV(2), 311–334 (2001)

  10. Waldi, R.: Wertehalbguppe und singularität einer ebenen algebraischen kurve. Dissertation. Regensburg (1972)

  11. Zariski, O.: The Moduli Problem for Plane Branches. University Lecture Series, AMS (2006)

Download references

Author information

Authors and Affiliations

  1. Universidade Federal Fluminense, Niterói, Brazil

    A. Hefez

  2. Universidade Estadual de Maringá, Maringá, Brazil

    M. E. Hernandes & M. E. Rodrigues Hernandes

Authors
  1. A. Hefez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. E. Hernandes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. E. Rodrigues Hernandes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Hefez.

Additional information

A. Hefez and M. E. Hernandes were partially supported by CNPq.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hefez, A., Hernandes, M.E. & Hernandes, M.E.R. The analytic classification of plane curves with two branches. Math. Z. 279, 509–520 (2015). https://doi.org/10.1007/s00209-014-1379-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-014-1379-2

Mathematics Subject Classification

Search

Navigation