Skip to main content
SpringerLink
Log in

On curves with all Weierstrass points of maximal weight

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

In this note we show that the few known examples of non-hyperelliptic complex algebraic curves all of whose Weierstrass points have maximal weight are the only ones.

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. R. D. M. Accola, Riemann surfaces, theta functions, and abelian automorphisms groups, Lecture Notes in Math. 483, Springer-Verlag (1975).

  2. R. D. M. Accola, Topics in the theory of Riemann surfaces, Lecture Notes in Math. 1595, Springer-Verlag (1994).

  3. E. Arbarello et al., Geometry of algebraic curves, Grundlehren Math. Wiss. 267, Springer-Verlag, Berlin-New York (1985).

  4. Ballico E., del Centina A.: Ramification points of double coverings of curves and Weierstrass points. Ann. di Mat. pura ed appl. (IV) 177, 293–313 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. del Centina A.: On certain remarkable curves of genus five. Indag. Math., N.S. 15, 339–346 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. P. G. Henn, Die Automorphismengruppen der algebraischen Funktionenkörper vom Geschlecht 3, PhD-thesis Univ. Heidelberg 1976.

  7. Horiuchi R.: Non-hyperelliptic Riemann surfaces of genus five all of whose Weierstrass points have maximal weight. Kodai Math. J. 30, 379–384 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kato T.: Non-hyperelliptic Weierstrass points of maximal weight. Math. Ann. 239, 141–147 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  9. T. Kato, Bi-elliptic Weierstrass points. Preprint 2009.

  10. KuribayashI A., Komiya K.: On Weierstrass points of non-hyperelliptic Riemann surfaces of genus 3. Hiroshima Math. J. 7, 743–768 (1977)

    MATH  MathSciNet  Google Scholar 

  11. T. Kato, K. Magaard, and H. Völklein, Bi-elliptic Weierstrass points on curves of genus 5, Preprint 2010.

  12. Kuribayashi A., Moriya R., Yoshida K.: On Weierstrass points. Bull. Fac. Sci. Eng. Chuo Univ. 20, 1–29 (1977)

    MathSciNet  Google Scholar 

  13. E. Lugert, Weierstrasspunkte kompakter Riemannscher Flächen vom Geschlecht 3, PhD-thesis Univ. Erlangen-Nürnberg (1981).

  14. Martens G., Schreyer F.-O.: Line bundles and syzygies of trigonal curves. Abh. Math. Sem. Univ. Hamburg 56, 169–189 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  15. K. Magaard et al., The locus of curves with prescribed automorphism group, RIMS, Kyoto Technical Report Ser., Communications on Arithmetic Fundamental Groups and Galois Theory, H. Nakamura (ed.) 21 pp., 2002.

  16. Teixidor i Bigas M.: For which Jacobi varieties is SingΘ reducible? J. Reine Angew. Math. 354, 141–149 (1984)

    MATH  MathSciNet  Google Scholar 

  17. Varley R.: Weddle’s surfaces, Humbert’s curves, and a certain 4-dimensional abelian variety. Amer. J. Math. 108, 931–952 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  18. A. M. Vermeulen, Weierstrass points of weight two on curves of genus 3, PhD-thesis Univ. Amsterdam 1983.

  19. A. Wiman, Über die algebraischen Kurven von den Geschlechtern 4, 5 und 6, welche eindeutige Transformationen in sich besitzen, Bihang till Kongl. Svenska Vetenskaps-Akademiens Handlingar 21, Afd. 1, Nr. 3 (1895).

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Seoul National University, 151-742, Seoul, South Korea

    Changho Keem

  2. Department Mathematik, Univ. Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054, Erlangen, Germany

    Gerriet Martens

Authors
  1. Changho Keem
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gerriet Martens
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gerriet Martens.

Additional information

The second author wants to thank Takao Kato for several discussions in Japan and, thereafter, by email; in fact, this paper could not have been written without his papers [8] and [9]. This author also wants to thank Helmut Völklein for discussions on Wiman’s curve (inducing Proposition 3.7).

The first author was supported in part by KRF 2009-0073208.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Keem, C., Martens, G. On curves with all Weierstrass points of maximal weight. Arch. Math. 94, 339–349 (2010). https://doi.org/10.1007/s00013-010-0108-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-010-0108-2

Mathematics Subject Classification (2010)

Keywords

Search

Navigation