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Classification up to equivalence of the holomorphic mappings of riemann surfaces of low genus

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Abstract

We give a complete classification up to equivalence of the holomorphic mappings of a Riemann surface of genus 3 onto a Riemann surface of genus 2. Also we establish that every Riemann surface of genus 3 has at most three holomorphic images of genus 2.

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References

  1. Hurwitz A., “Über Riemann’sche Flachen mit gegebenen Verzweigungspunkten,” Math. Ann., 39, 1–61 (1891).

    Article  MathSciNet  Google Scholar 

  2. Macbeath A. M., “On a theorem of Hurwitz,” Proc. Glasgow Math. Assoc., 5, 90–96 (1961).

    Article  MATH  MathSciNet  Google Scholar 

  3. De Franchis M., “Un teorema sulle involuzioni irrazionali,” Rend. Circ. Mat. Palermo, 36, 368 (1913).

    Article  MATH  Google Scholar 

  4. Howard A. and Sommese A. J., “On the theorem of de Franchis,” Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 10, No. 4, 429–436 (1983).

    MATH  MathSciNet  Google Scholar 

  5. Alzati A. and Pirola G. P., “Some remarks on the de Franchis theorem,” Ann. Univ. Ferrara Sez. VII, 36, 45–52 (1990).

    MATH  MathSciNet  Google Scholar 

  6. Tanabe M., “On rigidity of holomorphic maps of Riemann surfaces,” Osaka J. Math., 33, 485–496 (1996).

    MATH  MathSciNet  Google Scholar 

  7. Tanabe M., “A bound for the theorem of de Franchis,” Proc. Amer. Math. Soc., 127, 2289–2295 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  8. Tanabe M., “Holomorphic maps of Riemann surfaces and Weierstrass points,” Kodai Math. J., 28, No. 2, 423–429 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  9. Ito M. and Yamamoto H., “Holomorphic mappings between compact Riemann surfaces,” Proc. Edinburgh Math. Soc., 52, 109–126 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  10. Mednykh I. A., “On holomorphic maps between Riemann surfaces of genera three and two,” Dokl. Math., 424, No. 2, 29–31 (2009).

    Article  MathSciNet  Google Scholar 

  11. Fuertes Y., “Some bounds for the number of coincidences of morphisms between closed Riemann surfaces,” Israel J. Math., 109, No. 1, 1–12 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  12. Fuertes Y. and Gonzalez-Diez G., “On the number of coincidences of morphisms between closed Riemann surfaces,” Publ. Mat., 37, 339–353 (1993).

    MATH  MathSciNet  Google Scholar 

  13. Bandman T. M., “Surjective holomorphic mappings of projective manifolds,” Siberian Math. J., 22, 204–210 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  14. Imayoshi Y., “Generalizations of the de Franchis theorem,” Duke Math. J., 50, 393–408 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  15. Farkas H. M. and Kra I., Riemann Surfaces, Springer-Verlag, New York and Berlin (1980) (Graduate Texts Math.; 71).

    MATH  Google Scholar 

  16. Accola R. D. M., “On lifting the hyperelliptic involution,” Proc. Amer. Math. Soc., 122, No. 2, 341–347 (1994).

    MATH  MathSciNet  Google Scholar 

  17. Koebe P., “Über die Uniformisierung beliebiger analytischer Kurven,” Nachr. K. Ges.Wiss. Göttingen, 191–210; 633-669 (1907).

  18. Koebe P., “Über die Uniformisierung beliebiger analytischer Kurven Dritte Mitteilung,” Nachr. K. Ges.Wiss. Göttingen, 337–360 (1908).

  19. Satake I., “On a generalization of the notion of manifold,” Proc. Nat. Acad. Sci. USA, 42, 359–363 (1956).

    Article  MATH  MathSciNet  Google Scholar 

  20. Thurston W. P., The Geometry and Topology of 3-Manifolds, Princeton Univ., Princeton (1978).

    Google Scholar 

  21. Scott P., “The geometries of 3-manifolds,” Bull. London Math. Soc., 15, 401–487 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  22. Bolza O., “On binary sextics with linear transformations into themselves,” Amer. J. Math., 10, 47–60 (1888).

    Article  MathSciNet  Google Scholar 

  23. Magaard K., Shaska T., Shpectorov S., and Voelklein H., “The locus of curves with prescribed automorphism group,” RIMS Kyoto Series, Communications on Arithmetic Fundamental Groups, 6, 112–141 (2002).

    Google Scholar 

  24. Martens H., “A remark on Abel’s theorem and the mapping of linear series,” Comment. Math. Helv., 52, 557–559 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  25. Cardona G. and Quer J., “Curves of genus 2 with group of automorphisms isomorphic to D8 or D12,” Trans. Amer. Math. Soc., 359, No. 6, 2831–2849 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  26. Shaska T., “Determining the automorphism group of a hyperelliptic curve,” in: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, ACM Press, New York, 2003, pp. 248–254.

    Chapter  Google Scholar 

  27. Silhol R., “On some one-parameter families of genus 2 algebraic curves and half-twists,” Comment. Math. Helv., 82, 413–449 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  28. Kani E., “Bounds on the number of non-rational subfields of a function field,” Invent. Math., 85, 185–198 (1986).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Novosibirsk State University, Novosibirsk, Russia

    I. A. Mednykh

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  1. I. A. Mednykh
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Correspondence to I. A. Mednykh.

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Original Russian Text Copyright © 2010 Mednykh I. A.

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Mednykh, I.A. Classification up to equivalence of the holomorphic mappings of riemann surfaces of low genus. Sib Math J 51, 1091–1103 (2010). https://doi.org/10.1007/s11202-010-0107-3

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  • DOI: https://doi.org/10.1007/s11202-010-0107-3

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