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Results in Mathematics

, 74:194 | Cite as

The Cubo-Cubic Transformation and K3 Surfaces

  • Fabian ReedeEmail author
Article
  • 36 Downloads

Abstract

In this note we observe that the Cremona transformation in Oguiso’s example of Cremona isomorphic but not projectively equivalent quartic K3 surfaces in \({\mathbb {P}}^3\) is the classical cubo-cubic transformation of \({\mathbb {P}}^3\).

Keywords

Cremona transformations K3 surfaces determinantal hypersurfaces curves 

Mathematics Subject Classification

Primary 14E05 Secondary 14E07 14J28 14M12 

Notes

References

  1. 1.
    Beauville, A.: Determinantal hypersurfaces. Mich. Math. J. 48, 39–64 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cayley, A.: A memoir on quartic surfaces. Proc. Lond. Math. Soc. 3, 19–69 (1869/1871)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Coates, T., Corti, A., Galkin, S., Kasprzyk, A.: Quantum periods for 3-dimensional Fano manifolds. Geom. Topol. 20(1), 103–256 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dolgachev, I.V.: Classical Algebraic Geometry—A Modern View. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  5. 5.
    Ellingsrud, G.: Sur le schéma de Hilbert des variétés de codimension 2 dans \({ P}^{e}\) à cône de Cohen–Macaulay. Ann. Sci. École Norm. Sup. (4) 8(4), 423–431 (1975)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Festi, D., Garbagnati, A., van Geemen, B., van Luijk, R.: The Cayley–Oguiso automorphism of positive entropy on a K3 surface. J. Mod. Dyn. 7(1), 75–97 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Katz, S.: The cubo-cubic transformation of \({ P}^3\) is very special. Math. Z. 195(2), 255–257 (1987)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Matsumura, H., Monsky, P.: On the automorphisms of hypersurfaces. J. Math. Kyoto Univ. 3, 347–361 (1963/1964)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Noether, M.: Ueber die eindeutigen Raumtransformationen, insbesondere in ihrer Anwendung auf die Abbildung algebraischer Flächen. Math. Ann. 3(4), 547–580 (1871)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Oguiso, K.: Isomorphic quartic K3 surfaces in the view of Cremona and projective transformations. Taiwan. J. Math. 21(3), 671–688 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Semple, J.G., Roth, L.: Introduction to Algebraic Geometry. Clarendon Press, Oxford (1949)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für Algebraische GeometrieLeibniz Universität HannoverHannoverGermany

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