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Journal of Modern Dynamics (JMD)
 

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

Pages: 75 - 97, Issue 1, March 2013      doi:10.3934/jmd.2013.7.75

 
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Dino Festi - Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, 2333 Leiden, Netherlands (email)
Alice Garbagnati - Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italy (email)
Bert Van Geemen - Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italy (email)
Ronald Van Luijk - Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, 2333 Leiden, Netherlands (email)

Abstract: 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:  K3 surfaces, dynamics, automorphism, positive entropy.
Mathematics Subject Classification:  Primary: 14J28, 14J50, 37F10; Secondary: 32H50, 32J15, 32Q15.

Received: August 2012;      Available Online: May 2013.

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