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

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

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.
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.

[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.

[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.

[4]

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

[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.

[6]

A. Cayley, A memoir on quartic surfaces, Proc. London Math. Soc., 3 (1869-71), 19-69.

[7]

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

[8]

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

[9]

D. Festi, A. Garbagnati, B. van Geemen and R. van Luijk, Computations for Sections 4 and 5. Available from: http://www.math.leidenuniv.nl/~rvl/CayleyOguiso.

[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.

[11]

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

[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.

[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.

[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.

[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.

[16]

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

[17]

K. Oguiso, Free automorphisms of positive entropy on smooth Kähler surfaces, to appear in Adv. Stud. Pure Math., arXiv:1202.2637v3.

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[6]

A. Cayley, A memoir on quartic surfaces, Proc. London Math. Soc., 3 (1869-71), 19-69.

[7]

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

[8]

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

[9]

D. Festi, A. Garbagnati, B. van Geemen and R. van Luijk, Computations for Sections 4 and 5. Available from: http://www.math.leidenuniv.nl/~rvl/CayleyOguiso.

[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.

[11]

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

[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.

[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.

[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.

[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.

[16]

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

[17]

K. Oguiso, Free automorphisms of positive entropy on smooth Kähler surfaces, to appear in Adv. Stud. Pure Math., arXiv:1202.2637v3.

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[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.

[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.

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