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January  2013, 7(1): 75-97. doi: 10.3934/jmd.2013.7.75

The Cayley-Oguiso automorphism of positive entropy on a K3 surface

Dino Festi 1, , Alice Garbagnati 2, , Bert Van Geemen 2, and  Ronald Van Luijk 1,

1. 

Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, 2333 Leiden, Netherlands, Netherlands
2. 

Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italy, Italy

Received  August 2012 Published  May 2013

Recently Oguiso showed the existence of K3 surfaces that admit a fixed point free automorphism of positive entropy. The K3 surfaces used by Oguiso have a particular rank two Picard lattice. We show, using results of Beauville, that these surfaces are therefore determinantal quartic surfaces. Long ago, Cayley constructed an automorphism of such determinantal surfaces. We show that Cayley's automorphism coincides with Oguiso's free automorphism. We also exhibit an explicit example of a determinantal quartic whose Picard lattice has exactly rank two and for which we thus have an explicit description of the automorphism.
Keywords: automorphism, K3 surfaces, positive entropy., dynamics.
Mathematics Subject Classification: Primary: 14J28, 14J50, 37F10; Secondary: 32H50, 32J15, 32Q1.
Citation: Dino Festi, Alice Garbagnati, Bert Van Geemen, Ronald Van Luijk. The Cayley-Oguiso automorphism of positive entropy on a K3 surface. Journal of Modern Dynamics, 2013, 7 (1) : 75-97. doi: 10.3934/jmd.2013.7.75
References:
[1]

M. F. Atiyah and I. G. Macdonald, "Introduction to Commutative Algebra," Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.  Google Scholar

[2]

W. P. Barth, K. Hulek, C. A. M. Peters and A. Van de Ven, "Compact Complex Surfaces," Second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Folge, A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series, A Series of Modern Surveys in Mathematics], 4, Springer-Verlag, Berlin, 2004.  Google Scholar

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L. Bădescu, "Algebraic Surfaces," Translated from the 1981 Romanian original by Vladimir Maşek and revised by the author, Universitext, Springer-Verlag, New York, 2001.  Google Scholar

[4]

A. Beauville, Determinantal Hypersurfaces, Michigan Math. J., 48 (2000), 39-64. doi: 10.1307/mmj/1030132707.  Google Scholar

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S. Cantat, A. Chambert-Loir and V. Guedj, "Quelques Aspects des Systèmes Dynamiques Polynomiaux," Panoramas et Synthèses, 30, Société Mathématique de France, Paris, 2010.  Google Scholar

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I. Dolgachev, "Classical Algebraic Geometry: A Modern View," Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139084437.  Google Scholar

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W. Fulton, "Intersection Theory," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 2, Springer-Verlag, Berlin, 1984.  Google Scholar

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D. Festi, A. Garbagnati, B. van Geemen and R. van Luijk, Computations for Sections 4 and 5., Available from: \url{http://www.math.leidenuniv.nl/~rvl/CayleyOguiso}., ().   Google Scholar

[10]

A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math., 24 (1965), 231 pp.  Google Scholar

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R. Hartshorne, "Algebraic Geometry," Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

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Q. Liu, "Algebraic Geometry and Arithmetic Curves," Translated from the French by Reinie Erné, Oxford Graduate Texts in Mathematics, 6, Oxford Science Publications, Oxford University Press, Oxford, 2002.  Google Scholar

[13]

R. van Luijk, An elliptic K3 surface associated to Heron triangles, J. Number Theory, 123 (2007), 92-119. doi: 10.1016/j.jnt.2006.06.006.  Google Scholar

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R. van Luijk, K3 surfaces with Picard number one and infinitely many rational points, Algebra and Number Theory, 1 (2007), 1-15. doi: 10.2140/ant.2007.1.1.  Google Scholar

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W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.  Google Scholar

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J. S. Milne, "Étale Cohomology," Princeton Mathematical Series, 33, Princeton University Press, Princeton, N.J., 1980.  Google Scholar

[17]

K. Oguiso, Free automorphisms of positive entropy on smooth Kähler surfaces,, to appear in Adv. Stud. Pure Math., ().   Google Scholar

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T. G. Room, Self-transformations of determinantal quartic surfaces. I, Proc. London Math. Soc. (2), 51 (1950), 348-361. doi: 10.1112/plms/s2-51.5.348.  Google Scholar

[19]

T. G. Room, Self-transformations of determinantal quartic surfaces. II, Proc. London Math. Soc. (2), 51 (1950), 362-382. doi: 10.1112/plms/s2-51.5.362.  Google Scholar

[20]

T. G. Room, Self-transformations of determinantal quartic surfaces. III, Proc. London Math. Soc. (2), 51 (1950), 383-387. doi: 10.1112/plms/s2-51.5.383.  Google Scholar

[21]

T. G. Room, Self-transformations of determinantal quartic surfaces. IV, Proc. London Math. Soc. (2), 51 (1950), 388-400. doi: 10.1112/plms/s2-51.5.388.  Google Scholar

[22]

B. Saint-Donat, Projective models of K-3 surfaces, Amer. J. Math., 96 (1974), 602-639. doi: 10.2307/2373709.  Google Scholar

[23]

F. Schur, Über die durch collineare Grundgebilde erzeugten Curven und Flächen, Math. Ann., 18 (1881), 1-32. doi: 10.1007/BF01443653.  Google Scholar

[24]

V. Snyder and F. R. Sharpe, Certain quartic surfaces belonging to infinite discontinuous Cremonian groups, Trans. Amer. Math. Soc., 16 (1915), 62-70. doi: 10.1090/S0002-9947-1915-1501000-2.  Google Scholar

[25]

J. T. Tate, Algebraic cycles and poles of zeta functions, in "Arithmetical Algebraic Geometry" (Proc. Conf. Purdue Univ., 1963), Harper & Row, New York, (1965), 93-110.  Google Scholar

show all references

References:
[1]

M. F. Atiyah and I. G. Macdonald, "Introduction to Commutative Algebra," Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.  Google Scholar

[2]

W. P. Barth, K. Hulek, C. A. M. Peters and A. Van de Ven, "Compact Complex Surfaces," Second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Folge, A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series, A Series of Modern Surveys in Mathematics], 4, Springer-Verlag, Berlin, 2004.  Google Scholar

[3]

L. Bădescu, "Algebraic Surfaces," Translated from the 1981 Romanian original by Vladimir Maşek and revised by the author, Universitext, Springer-Verlag, New York, 2001.  Google Scholar

[4]

A. Beauville, Determinantal Hypersurfaces, Michigan Math. J., 48 (2000), 39-64. doi: 10.1307/mmj/1030132707.  Google Scholar

[5]

S. Cantat, A. Chambert-Loir and V. Guedj, "Quelques Aspects des Systèmes Dynamiques Polynomiaux," Panoramas et Synthèses, 30, Société Mathématique de France, Paris, 2010.  Google Scholar

[6]

A. Cayley, A memoir on quartic surfaces,, Proc. London Math. Soc., 3 (): 1869.   Google Scholar

[7]

I. Dolgachev, "Classical Algebraic Geometry: A Modern View," Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139084437.  Google Scholar

[8]

W. Fulton, "Intersection Theory," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 2, Springer-Verlag, Berlin, 1984.  Google Scholar

[9]

D. Festi, A. Garbagnati, B. van Geemen and R. van Luijk, Computations for Sections 4 and 5., Available from: \url{http://www.math.leidenuniv.nl/~rvl/CayleyOguiso}., ().   Google Scholar

[10]

A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math., 24 (1965), 231 pp.  Google Scholar

[11]

R. Hartshorne, "Algebraic Geometry," Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[12]

Q. Liu, "Algebraic Geometry and Arithmetic Curves," Translated from the French by Reinie Erné, Oxford Graduate Texts in Mathematics, 6, Oxford Science Publications, Oxford University Press, Oxford, 2002.  Google Scholar

[13]

R. van Luijk, An elliptic K3 surface associated to Heron triangles, J. Number Theory, 123 (2007), 92-119. doi: 10.1016/j.jnt.2006.06.006.  Google Scholar

[14]

R. van Luijk, K3 surfaces with Picard number one and infinitely many rational points, Algebra and Number Theory, 1 (2007), 1-15. doi: 10.2140/ant.2007.1.1.  Google Scholar

[15]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.  Google Scholar

[16]

J. S. Milne, "Étale Cohomology," Princeton Mathematical Series, 33, Princeton University Press, Princeton, N.J., 1980.  Google Scholar

[17]

K. Oguiso, Free automorphisms of positive entropy on smooth Kähler surfaces,, to appear in Adv. Stud. Pure Math., ().   Google Scholar

[18]

T. G. Room, Self-transformations of determinantal quartic surfaces. I, Proc. London Math. Soc. (2), 51 (1950), 348-361. doi: 10.1112/plms/s2-51.5.348.  Google Scholar

[19]

T. G. Room, Self-transformations of determinantal quartic surfaces. II, Proc. London Math. Soc. (2), 51 (1950), 362-382. doi: 10.1112/plms/s2-51.5.362.  Google Scholar

[20]

T. G. Room, Self-transformations of determinantal quartic surfaces. III, Proc. London Math. Soc. (2), 51 (1950), 383-387. doi: 10.1112/plms/s2-51.5.383.  Google Scholar

[21]

T. G. Room, Self-transformations of determinantal quartic surfaces. IV, Proc. London Math. Soc. (2), 51 (1950), 388-400. doi: 10.1112/plms/s2-51.5.388.  Google Scholar

[22]

B. Saint-Donat, Projective models of K-3 surfaces, Amer. J. Math., 96 (1974), 602-639. doi: 10.2307/2373709.  Google Scholar

[23]

F. Schur, Über die durch collineare Grundgebilde erzeugten Curven und Flächen, Math. Ann., 18 (1881), 1-32. doi: 10.1007/BF01443653.  Google Scholar

[24]

V. Snyder and F. R. Sharpe, Certain quartic surfaces belonging to infinite discontinuous Cremonian groups, Trans. Amer. Math. Soc., 16 (1915), 62-70. doi: 10.1090/S0002-9947-1915-1501000-2.  Google Scholar

[25]

J. T. Tate, Algebraic cycles and poles of zeta functions, in "Arithmetical Algebraic Geometry" (Proc. Conf. Purdue Univ., 1963), Harper & Row, New York, (1965), 93-110.  Google Scholar

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