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

Dino Festi,Ronald Van Luijk,Bert Van Geemen,Alice Garbagnati

doi:10.3934/jmd.2013.7.75

Abstract

Recently Oguiso showed the existence of K3 surfaces that admit a fixedpoint free automorphism of positive entropy. The K3 surfaces used byOguiso have a particular rank two Picard lattice. We show, usingresults of Beauville, that these surfaces are therefore determinantalquartic surfaces. Long ago, Cayley constructed an automorphism ofsuch determinantal surfaces. We show that Cayley's automorphismcoincides with Oguiso's free automorphism. We also exhibit anexplicit example of a determinantal quartic whose Picard lattice hasexactly rank two and for which we thus have an explicit description ofthe automorphism.

Highlights

Oguiso showed the existence of K3 surfaces that admit a fixed point free automorphism of positive entropy
Keiji Oguiso showed that there exist projective K3 surfaces S with a fixed point free automorphism g of positive entropy, i.e. g∗ has at least one eigenvalue λ of absolute value |λ| > 1 on H2(S, C)
The aim of this paper is to provide a general method for constructing such quartic surfaces in P3 and to describe an algorithm for finding the automorphism

Summary

The general constructions

As η2(1 − 2η)/5 = (−1 − 3η)/5, the generator x → η2x of the isometry group induces the map x → −x on the subgroup Z/5Z of N ∗/N. Let L be the line bundle on S defined by an ample divisor class D. The zero locus of a global section t of D3 is mapped to a curve in S0 This curve is not the (complete) intersection of S0 with another surface (of degree d) in P3, since such an intersection has class dD0 = d, whereas D3 = η6 = 5 + 8η. We were able to find the degree 18 polynomials in the specific example in Section 4, see Section 4.4

Determinantal quartic surfaces

Cayley’s description of the automorphism

An explicit example

The explicit example and periodic points