American Institute of Mathematical Sciences

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
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