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manuscripta mathematica

, Volume 124, Issue 4, pp 459–480 | Cite as

Fixed points of automorphisms of real algebraic curves

  • Jean-Philippe Monnier
Article
  • 45 Downloads

Abstract

We bound the number of fixed points of an automorphism of a real curve in terms of the genus and the number of connected components of the real part of the curve. Using this bound, we derive some consequences concerning the maximum order of an automorphism and the maximum order of an abelian group of automorphisms of a real curve. We also bound the full group of automorphisms of a real hyperelliptic curve.

Mathematics Subject Classification (2000)

14H37 14P25 14P99 

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References

  1. 1.
    Bujalance, E., Cirre, F.-J., Gamboa, J.-M., Gromadzki, G.: Symmetry types of Hyperelliptic Riemann Surfaces. Mèm. Soc. Math. Fr. 86, (2001)Google Scholar
  2. 2.
    Bujalance, E., Etayo, E., Gamboa, J.-M., Gromadzki, G.: Automorphism groups of compact bordered Klein surfaces. In: Lecture Notes in Mathematics, p. 1439. Springer, Berlin (1990)Google Scholar
  3. 3.
    Farkas H.M. and Kra I. (1980). Riemann Surfaces. Springer, Heidelberg zbMATHGoogle Scholar
  4. 4.
    Gross G.H. and Harris J. (1981). Real algebraic curves. Ann. Sci. Ecole Norm. Sup. 14(4): 157–182 zbMATHMathSciNetGoogle Scholar
  5. 5.
    Hurwitz A. (1893). Uber algebraische Gebilde mit eindeutingen Transformationen in sich. Math. Annalen 41: 403–442 CrossRefMathSciNetGoogle Scholar
  6. 6.
    Knight J.T. (1969). Riemman surfaces of field extensions. Proc. Cambridge Philos. Soc. 65: 635–650 zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Krull W. and Neukirch J. (1971). Die Struktur der absoluten Galoisgruppe über dem Körper \({\mathbb{R}}(t)\) Math. Ann. 193: 197–209 zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    May C.L. (1975). Automorphisms of compact Klein surfaces with boundary. Pacific J. Math. 59: 199–210 zbMATHMathSciNetGoogle Scholar
  9. 9.
    May C.L. (1977). Cyclic automorphisms groups of compact bordered Klein surfaces. Houston J. Math. 3: 395–405 zbMATHMathSciNetGoogle Scholar
  10. 10.
    Monnier J.P. (2003). Divisors on real curves. Adv. Geom 3: 339–360 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Natanzon S. (1990). Klein surfaces. Uspekhi Mat. Nauk transl. Russian Math. Surv. 45(6): 53–108 zbMATHMathSciNetGoogle Scholar
  12. 12.
    Natanzon S. (1989). Finite groups of homeomorphisms of surfaces and real forms of complex algebraic curves. Transl. Trans. Moscow Math. Soc. 51: 1–51 MathSciNetGoogle Scholar
  13. 13.
    Pardini R. (1991). Abelian covers of algebraic varieties. J. Reine Angew. Math. 417: 191–213 zbMATHMathSciNetGoogle Scholar
  14. 14.
    Wiman, A.: Uber die hyperelliptischen Curven und diejenigen vom Geschlechte p = 3, welche eindeutingen transformationnen in such zulassen. Bihang Till Kongl. Svenska Vetenskaps-Akademiens Hadlingar, Stockholm, pp. 1895–1896Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité d’AngersAngers cedex 01France

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