American Institute of Mathematical Sciences

Advanced
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., (1969).   Google Scholar

[2]

W. P. Barth, K. Hulek, C. A. M. Peters and A. Van de Ven, "Compact Complex Surfaces,", Second edition, 4 (2004).   Google Scholar

[3]

L. Bădescu, "Algebraic Surfaces,", Translated from the 1981 Romanian original by Vladimir Maşek and revised by the author, (1981).   Google Scholar

[4]

A. Beauville, Determinantal Hypersurfaces,, Michigan Math. J., 48 (2000), 39.  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 (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, (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 (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).   Google Scholar

[11]

R. Hartshorne, "Algebraic Geometry,", Graduate Texts in Mathematics, (1977).   Google Scholar

[12]

Q. Liu, "Algebraic Geometry and Arithmetic Curves,", Translated from the French by Reinie Erné, 6 (2002).   Google Scholar

[13]

R. van Luijk, An elliptic K3 surface associated to Heron triangles,, J. Number Theory, 123 (2007), 92.  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.  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.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[16]

J. S. Milne, "Étale Cohomology,", Princeton Mathematical Series, 33 (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.  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.  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.  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.  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.  doi: 10.2307/2373709.  Google Scholar

[23]

F. Schur, Über die durch collineare Grundgebilde erzeugten Curven und Flächen,, Math. Ann., 18 (1881), 1.  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.  doi: 10.1090/S0002-9947-1915-1501000-2.  Google Scholar

[25]

J. T. Tate, Algebraic cycles and poles of zeta functions,, in, (1965), 93.   Google Scholar

show all references

References:
[1]

M. F. Atiyah and I. G. Macdonald, "Introduction to Commutative Algebra,", Addison-Wesley Publishing Co., (1969).   Google Scholar

[2]

W. P. Barth, K. Hulek, C. A. M. Peters and A. Van de Ven, "Compact Complex Surfaces,", Second edition, 4 (2004).   Google Scholar

[3]

L. Bădescu, "Algebraic Surfaces,", Translated from the 1981 Romanian original by Vladimir Maşek and revised by the author, (1981).   Google Scholar

[4]

A. Beauville, Determinantal Hypersurfaces,, Michigan Math. J., 48 (2000), 39.  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 (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, (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 (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).   Google Scholar

[11]

R. Hartshorne, "Algebraic Geometry,", Graduate Texts in Mathematics, (1977).   Google Scholar

[12]

Q. Liu, "Algebraic Geometry and Arithmetic Curves,", Translated from the French by Reinie Erné, 6 (2002).   Google Scholar

[13]

R. van Luijk, An elliptic K3 surface associated to Heron triangles,, J. Number Theory, 123 (2007), 92.  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.  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.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[16]

J. S. Milne, "Étale Cohomology,", Princeton Mathematical Series, 33 (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.  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.  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.  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.  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.  doi: 10.2307/2373709.  Google Scholar

[23]

F. Schur, Über die durch collineare Grundgebilde erzeugten Curven und Flächen,, Math. Ann., 18 (1881), 1.  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.  doi: 10.1090/S0002-9947-1915-1501000-2.  Google Scholar

[25]

J. T. Tate, Algebraic cycles and poles of zeta functions,, in, (1965), 93.   Google Scholar

[1]

Günther J. Wirsching. On the problem of positive predecessor density in $3n+1$ dynamics. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 771-787. doi: 10.3934/dcds.2003.9.771
[2]

Marcelo R. R. Alves. Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds. Journal of Modern Dynamics, 2016, 10: 497-509. doi: 10.3934/jmd.2016.10.497
[3]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147
[4]

Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75
[5]

Seung Won Kim, P. Christopher Staecker. Dynamics of random selfmaps of surfaces with boundary. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 599-611. doi: 10.3934/dcds.2014.34.599
[6]

J. L. Barbosa, L. Birbrair, M. do Carmo, A. Fernandes. Globally subanalytic CMC surfaces in $\mathbb{R}^3$. Electronic Research Announcements, 2014, 21: 186-192. doi: 10.3934/era.2014.21.186
[7]

Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581
[8]

François Ledrappier. Erratum: On Omri Sarig's work on the dynamics of surfaces. Journal of Modern Dynamics, 2015, 9: 355-355. doi: 10.3934/jmd.2015.9.355
[9]

François Ledrappier. On Omri Sarig's work on the dynamics on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 15-24. doi: 10.3934/jmd.2014.8.15
[10]

François Gay-Balmaz, Cesare Tronci, Cornelia Vizman. Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups. Journal of Geometric Mechanics, 2013, 5 (1) : 39-84. doi: 10.3934/jgm.2013.5.39
[11]

Dwayne Chambers, Erica Flapan, John D. O'Brien. Topological symmetry groups of $K_{4r+3}$. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1401-1411. doi: 10.3934/dcdss.2011.4.1401
[12]

Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006
[13]

Dong Chen. Positive metric entropy in nondegenerate nearly integrable systems. Journal of Modern Dynamics, 2017, 11: 43-56. doi: 10.3934/jmd.2017003
[14]

Alejo Barrio Blaya, Víctor Jiménez López. On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 433-466. doi: 10.3934/dcds.2012.32.433
[15]

Jorge Sotomayor, Ronaldo Garcia. Codimension two umbilic points on surfaces immersed in $R^3$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 293-308. doi: 10.3934/dcds.2007.17.293
[16]

Suleyman Tek. Some classes of surfaces in $\mathbb{R}^3$ and $\M_3$ arising from soliton theory and a variational principle. Conference Publications, 2009, 2009 (Special) : 761-770. doi: 10.3934/proc.2009.2009.761
[17]

Anatole Katok. Fifty years of entropy in dynamics: 1958--2007. Journal of Modern Dynamics, 2007, 1 (4) : 545-596. doi: 10.3934/jmd.2007.1.545
[18]

Fryderyk Falniowski, Marcin Kulczycki, Dominik Kwietniak, Jian Li. Two results on entropy, chaos and independence in symbolic dynamics. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3487-3505. doi: 10.3934/dcdsb.2015.20.3487
[19]

Matías Navarro, Federico Sánchez-Bringas. Dynamics of principal configurations near umbilics for surfaces in $mathbb(R)^4$. Conference Publications, 2003, 2003 (Special) : 664-671. doi: 10.3934/proc.2003.2003.664
[20]

Boris Kruglikov, Martin Rypdal. A piece-wise affine contracting map with positive entropy. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 393-394. doi: 10.3934/dcds.2006.16.393

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences