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.
The Cayley-Oguiso automorphism of positive entropy on a K3 surface / D. Festi, A. Garbagnati, B. van Geemen, R. van Luijk. - In: JOURNAL OF MODERN DYNAMICS. - ISSN 1930-5311. - 1:1(2013 Mar), pp. 75-97. [10.3934/jmd.2013.7.75]
The Cayley-Oguiso automorphism of positive entropy on a K3 surface
A. GarbagnatiSecondo
;B. van GeemenPenultimo
;
2013
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.Pubblicazioni consigliate
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