Skip to main content
SpringerLink
Log in

Coble surfaces with finite automorphism group

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

We classify Coble surfaces with finite automorphism group in characteristic \(p\ge 0, p\ne 2\). There are exactly 9 isomorphism classes of such surfaces.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23

Similar content being viewed by others

References

  1. Artin, M.: Supersingular \(K3\) surfaces. Ann. Sci. École Norm. Sup. 7, 543–567 (1974)

    Article  MathSciNet  Google Scholar 

  2. Bombieri, E., Mumford, D.: Enriques’ classification of surfaces in char. \(p\), III. Invent. math. 35, 197–232 (1976)

    Article  MathSciNet  Google Scholar 

  3. Coble, A.: The ten nodes of the rational sextic and of the Cayley symmetroid. Am. J. Math. 41, 243–265 (1919)

    Article  MathSciNet  Google Scholar 

  4. Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Clarendon Press, Oxford (1985)

    MATH  Google Scholar 

  5. Cossec, F.: On the Picard group of Enriques surfaces. Math. Ann. 271, 577–600 (1985)

    Article  MathSciNet  Google Scholar 

  6. Cossec, F., Dolgachev, I., Liedtke, C.: Enriques surfaces I, http://www.math.lsa.umich.edu/idolga/EnriquesTwo.pdf

  7. Dolgachev, I., Kondō, S.: Enriques surfaces II, http://www.math.lsa.umich.edu/~idolga/EnriquesTwo.pdf

  8. Dolgachev, I.: On automorphisms of Enriques surfaces. Invent. Math. 76, 163–177 (1984)

    Article  MathSciNet  Google Scholar 

  9. Dolgachev, I., Zhang, D.-Q.: Coble rational surfaces. Am. J. Math. 123, 79–114 (2001)

    Article  MathSciNet  Google Scholar 

  10. Ito, H.: The Mordell-Weil groups of unirational quasi-elliptic surfaces in characteristic \(3\). Math. Z. 211, 1–39 (1992)

    Article  MathSciNet  Google Scholar 

  11. Ito, H.: On extremal elliptic surfaces in characteristic \(2\) and \(3\). Hiroshima Math. J. 32, 179–188 (2002)

    Article  MathSciNet  Google Scholar 

  12. Katsura, T., Kondō, S.: Rational curves on the supersingular \(K3\) surface with Artin invariant \(1\) in characteristic \(3\). J. Algebra 352, 299–321 (2012)

    Article  MathSciNet  Google Scholar 

  13. Katsura, T., Kondō, S., Martin, G.: Classification of Enriques surfaces with finite automorphism group in characteristic \(2\). Algebraic Geometry 7, 390–459 (2020)

    MathSciNet  MATH  Google Scholar 

  14. Kondō, S.: Enriques surfaces with finite automorphism groups. Jpn. J. Math. 12, 191–282 (1986)

    Article  MathSciNet  Google Scholar 

  15. Kondō, S.: Classification of Enriques surfaces covered by the supersingular \(K3\) surface with Artin invariant \(1\) in characteristic \(2\). J. Math. Soc. Jpn 73, 301–328 (2021)

    Article  MathSciNet  Google Scholar 

  16. Kondō, S.: \(K3\) surfaces, Tracts in Mathematics, European Math. Soc. (2020)

  17. Martin, G.: Enriques surfaces with finite automorphism group in positive characteristic. Algebraic Geometry 6, 592–649 (2019)

    Article  MathSciNet  Google Scholar 

  18. Morrison, D.: Semistable degenerations of Enriques’ and hyperelliptic surfaces. Duke Math. J. 48, 197–249 (1981)

    Article  MathSciNet  Google Scholar 

  19. Mukai, S., Ohashi, H.: Finite groups of automorphisms of Enriques surfaces and the Mathieu group \(M_{12}\), arXiv:1410.7535.2014

  20. Mukai, S., Ohashi, H.: Automorphisms of two rational surfaces in characteristic \(3\), in preparation

  21. Nikulin, V.: Integral symmetric bilinear forms and its applications. Math. USSR Izv. 14, 103–167 (1980)

    Article  Google Scholar 

  22. Nikulin, V.V.: Quotient-groups of groups of automorphisms of hyperbolic forms modulo subgroups generated by \(2\)-reflections, algebraic geometric applications. J. Soviet Math. 22, 1401–1476 (1983)

    Article  Google Scholar 

  23. Nikulin, V.V.: On a description of the automorphism groups of Enriques surfaces. Soviet. Math. Dokl. 30, 282–285 (1984)

    MATH  Google Scholar 

  24. Schütt, M., Shioda, T.: Mordell-Weil Lattices. Springer, Berlin (2019)

    Book  Google Scholar 

  25. Shioda, T.: Supersingular \(K3\) Surfaces, Algebraic Geometry, Copenhagen 1978, pp. 564–591, Lecture Notes in Mathematics 732, Springer, (1979)

  26. Vinberg, E.B.: Some arithmetic discrete groups in Lobachevskii spaces, in Discrete subgroups of Lie groups and applications to Moduli. Tata-Oxford, pp. 323–348, (1975)

Download references

Acknowledgements

The author thanks Igor Dolgachev and Shigeru Mukai for teaching him a beauty of Coble surfaces and for stimulating discussions and advices, and the referee for the careful reading of the manuscript, for pointing out errors and for many useful suggestions.

Author information

Authors and Affiliations

  1. Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan

    Shigeyuki Kondō

Authors
  1. Shigeyuki Kondō
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shigeyuki Kondō.

Additional information

In memory of Professor Ernest Vinberg.

Publisher's Note

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

Research of the author is partially supported by Grant-in-Aid for Scientific Research (A) No.20H00112.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kondō, S. Coble surfaces with finite automorphism group. Rend. Circ. Mat. Palermo, II. Ser 71, 829–864 (2022). https://doi.org/10.1007/s12215-021-00646-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-021-00646-2

Keywords

Mathematics Subject Classification

Search

Navigation