Classification of Rational Unicuspidal Projective Curves whose Singularities Have one Puiseux Pair

  • Javier Fernández de Bobadilla
  • Ignacio Luengo
  • Alejandro Melle Hernández
  • Andras Némethi
Conference paper
Part of the Trends in Mathematics book series (TM)


It is a very old and interesting open problem to characterize those collections of embedded topological types of local plane curve singularities which may appear as singularities of a projective plane curve C of degree d. The goal of the present article is to give a complete (topological) classification of those cases when C is rational and it has a unique singularity which is locally irreducible (i.e., C is unicuspidal) with one Puiseux pair.


Cuspidal rational plane curves logarithmic Kodaira dimension 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Dimca, A.: Singularities and Topology of Hypersurfaces, Universitext, Springer-Verlag, New York, 1992.zbMATHGoogle Scholar
  2. [2]
    Fernández de Bobadilla J., Luengo I., Melle-Hernández A., Némethi A.: On rational cuspidal projective plane curves, Proc. London Math. Soc. (3), 92, (2006), 99–138.MathSciNetGoogle Scholar
  3. [3]
    Fernández de Bobadilla J., Luengo I., Melle-Hernández A., Némethi A.: On rational cuspidal curves, open surfaces and local singularities, arXiv.math.AG/0604421.Google Scholar
  4. [4]
    Fujita, T.: On the topology of non-complete algebraic surfaces, J. Fac. Sci. Univ. Tokyo (Ser1A), 29 (1982), 503–566.Google Scholar
  5. [5]
    Kashiwara, H.: Fonctions rationelles de type (0,1) sur le plan projectif complexe, Osaka J. Math., 24 (1987), 521–577.MathSciNetGoogle Scholar
  6. [6]
    Matsuoka, T. and Sakai, F.: The degree of of rational cuspidal curves, Math. Ann., 285 (1989), 233–247.CrossRefMathSciNetGoogle Scholar
  7. [7]
    Miyanishi, M. and Sugie, T.: On a projective plane curve whose complement has logarithmic Kodaira dimension −∞, Osaka J. Math., 18 (1981), 1–11.MathSciNetGoogle Scholar
  8. [8]
    Namba, M.: Geometry of projective algebraic curves. Monographs and Textbooks in Pure and Applied Mathematics, 88 Marcel Dekker, Inc., New York, 1984.zbMATHGoogle Scholar
  9. [9]
    Orevkov, S. Yu.: On rational cuspidal curves, I. Sharp estimate for degree via multiplicities, Math. Ann. 324 (2002), 657–673.CrossRefMathSciNetGoogle Scholar
  10. [10]
    Tono, K.: On rational unicuspidal plane curves with logarithmic Kodaira dimension one, preprint.Google Scholar
  11. [11]
    Tsunoda, Sh.: The complements of projective plane curves, RIMS-Kôkyûroku, 446 (1981), 48–56.Google Scholar
  12. [12]
    Vajda, S.: Fibonacci and Lucas numbers, and the Golden Section: Theory and Applications, Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press New York, 1989.zbMATHGoogle Scholar
  13. [13]
    Varchenko, A.N.: On the change of discrete characteristics of critical points of functions under deformations, Uspekhi Mat. Nauk, 38:5 (1985), 126–127.Google Scholar
  14. [14]
    Varchenko, A.N.: Asymptotics of integrals and Hodge structures. Science reviews: current problems in mathematics 1983. 22, 130–166; J. Sov. Math. 27 (1984).MathSciNetGoogle Scholar
  15. [15]
    Wall, C.T.C.: Singular Points of Plane Curves, London Math. Soc. Student Texts 63, Cambridge University Press, 2004.Google Scholar
  16. [16]
    Yoshihara, Y.: Rational curves with one cusp (in Japanese), Sugaku, 40 (1988), 269–271.MathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Javier Fernández de Bobadilla
    • 1
  • Ignacio Luengo
    • 2
  • Alejandro Melle Hernández
    • 2
  • Andras Némethi
    • 3
    • 4
  1. 1.Department of MathematicsUniversity of UtrechtUtrechtThe Netherlands
  2. 2.Facultad de MatemáticasUniversidad ComplutenseMadridSpain
  3. 3.Department of MathematicsOhio State UniversityColumbusUSA
  4. 4.Rényi Institute of MathematicsBudapestHungary

Personalised recommendations