Skip to main content
Log in

Dihedral Groups and Smooth Plane Curves with Many Quasi-Galois Points

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

A point p in projective plane is said to be quasi-Galois for a plane curve if the curve admits a non-trivial birational transformation which preserves the fibers of the projection \(\pi _p\) from the point p. A number of quasi-Galois points for smooth plane curves of degree d are studied. In this paper, we will study the relationship between dihedral groups and smooth plane curves with many quasi-Galois points.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Badr, E., Bars, F.: Automorphism groups of non-singular plane curves of degree 5. Commun. Algebr. 44, 4327–4340 (2016)

    Article  Google Scholar 

  2. Badr, E., Bars, F.: Non-singular plane curves with an element of large order in its automorphism group. Int. J. Algebr. Comput. 26, 399–434 (2016)

    Article  MathSciNet  Google Scholar 

  3. Blichfeldt, H.: Finite Collineation Groups: With an Introduction to the Theory of Groups of Operators and Substitution Groups. University of Chicago Press, Chicago (1917)

    Google Scholar 

  4. Fukasawa, S., Miura, K., Takahashi, T.: Quasi-Galois points, I: Automorphism groups of plane curves. Tohoku Math. J. 71(4), 487–494 (2019)

    Article  MathSciNet  Google Scholar 

  5. Harui, T.: Automorphism groups of smooth plane curves. Kodai Math. J. 42(2), 308–331 (2019)

    Article  MathSciNet  Google Scholar 

  6. Harui, T.: Smooth plane curves whose automorphism group is primitive, preprint

  7. Harui, T., Kato, T., Komeda, J., Ohbuchi, A.: Quotient curves of smooth plane curves with automorphisms. Kodai Math. J. 33(1), 164–172 (2010)

    Article  MathSciNet  Google Scholar 

  8. Hayashi, T.: Smooth plane curves with freely acting finite groups. Vietnam J. Math. (2020). https://doi.org/10.1007/s10013-020-00398-z

    Article  Google Scholar 

  9. Hayashi, T.: Linear automorphisms of hypersurfaces giving Galois points, to appear in Bull. Korean Math. Soc, arXiv:2101.04797

  10. Hayashi, T.: Orders of automorphisms of smooth plane curves for the automorphism groups to be cyclic, to appear. Math. Arabian J. (2021). https://doi.org/10.1007/s40065-021-00321-5

  11. Henn, P.:Die Automorphismengruppen dar algebraischen Functionenkorper vom Geschlecht 3, Inagural-dissertation, Heidelberg, 1976

  12. Kuribayashi, A., Komiya, K.: On Weierstrass points of non-hyperelliptic compact Riemann surfaces of genus three. Hiroshima Math. J. 7, 743–786 (1977)

    Article  MathSciNet  Google Scholar 

  13. Miura, K., Yoshihara, H.: Field theory for function fields of plane quartic curves. J. Algebr. 226, 283–294 (2000)

    Article  MathSciNet  Google Scholar 

  14. Namba, M.: Geometry of projective algebraic curves. Marcel Dekker, NY (1984)

    MATH  Google Scholar 

  15. Tzermias, P.: The group of automorphisms of the Fermat curve. J. Number Theory 53(1), 173–178 (1995)

    Article  MathSciNet  Google Scholar 

  16. Yoshihara, H.: Function field theory of plane curves by dual curves. J. Algebr. 239, 340–355 (2001)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Taro Hayashi.

Additional information

Communicated by Rosihan M. Ali.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hayashi, T. Dihedral Groups and Smooth Plane Curves with Many Quasi-Galois Points. Bull. Malays. Math. Sci. Soc. 44, 4251–4267 (2021). https://doi.org/10.1007/s40840-021-01164-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-021-01164-1

Keywords

Mathematics Subject Classification

Navigation