Skip to main content
SpringerLink
Log in

On smooth separatrices of foliations on singular surfaces

  • Published:
Bulletin of the Brazilian Mathematical Society, New Series Aims and scope Submit manuscript

Abstract

We consider foliations on a germ of an isolated surface singularity. We give criteria for the existence of smooth separatrices in terms of the maximal cycle of a desingularization of the surface. We apply these criteria for some examples of quasi-homogeneous singularities, in particular rational double points; moreover, for these singularities we show for which cases there exist a smooth separatrix for foliations defined by sufficiently general derivations.

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. A.G. Aleksandrov. Cohomology of quasi-homogeneous complete intersection. Math. USSR Izv., 26 (1986), 437–477.

    Article  MATH  Google Scholar 

  2. M. Artin. On isolated rational singularities of surfaces. Amer. J.Math., 88 (1966), 129–136.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Artin. Some numerical criteria for contractability of curves on algebraic surfaces. Amer. J. Math., 84 (1962), 485–496.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Brunella. BirationalGeometry of Foliations. First Latinoamerican Congress of Mathematics, IMPA, (2001).

    Google Scholar 

  5. C. Camacho. Quadratic Forms and Holomorphic Foliations on Singular Surfaces. Math. Ann., 282 (1988), 167–184.

    Article  MathSciNet  Google Scholar 

  6. C. Camacho and P. Sad. Invariant varieties through singularities of homolorphic vector fields. Ann. of Math., 115 (1982), 579–595.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Durfee. Fifteen characterization of rational double points and simple critical points. L’Enseignement Mathématique, 25 (1979).

    Google Scholar 

  8. G. Gonzalez-Sprinberg. Résolution de Nash de points doubles rationnels. Ann. Inst. Fourie., 32(2) (1982), 111–178.

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Gonzalez-Sprinberg and M. Lejeune-Jalabert. Courbes lisses sur les singularités de surfaces. C.R. Acad. Sci. Paris, t. 318, Série I Math., (1994), 653–656.

    Google Scholar 

  10. G. Gonzalez-Sprinberg and J.-L. Verdier. Construction géométrique de la correspondance de McKay. Ann. Sci. Ecole Norm. Sup., 16 (1983), 409–449.

    MATH  MathSciNet  Google Scholar 

  11. H. Pinkham. Singularités de Klein. Séminaire sur le singularités des surfaces, Springer LNM, 777 (1980).

    Google Scholar 

  12. A. Seidenberg. Reduction of Singularitiesof the differential equation Ady- Bdx. Amer. J. Math., 90 (1968), 248–269.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Wahl. Derivations, automorphisms and deformations of quasi-homogeneous singularities. Proc. Symp. Pure Math., 40 (part 2) (1983), 613–624.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centro de Matemática, Facultad de Ciencias, UdelaR, Montevideo, Uruguay

    Gerardo Gonzalez-Sprinberg & Ivan Pan

  2. Institut Fourier, Université de Grenoble I, Grenoble, France

    Gerardo Gonzalez-Sprinberg

Authors
  1. Gerardo Gonzalez-Sprinberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ivan Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivan Pan.

Additional information

Partially supported by IFUM and CSIC-UdelaR.

Partially supported by Agencia Nacional de Investigadores of Uruguay and CSIC-UdelaR.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gonzalez-Sprinberg, G., Pan, I. On smooth separatrices of foliations on singular surfaces. Bull Braz Math Soc, New Series 46, 1–21 (2015). https://doi.org/10.1007/s00574-015-0082-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-015-0082-1

Keywords

Mathematical subject classification

Search

Navigation