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

TL;DR

This paper demonstrates that certain K3 surfaces with a specific Picard lattice admit a known automorphism of positive entropy, linking Oguiso's and Cayley's automorphisms and providing explicit examples.

Contribution

It establishes the equivalence of Oguiso's automorphism with Cayley's classical automorphism on determinantal quartic K3 surfaces and provides explicit examples with rank two Picard lattice.

Findings

Oguiso's automorphism coincides with Cayley's automorphism.

Determinantal quartic surfaces can have automorphisms of positive entropy.

Explicit examples of such K3 surfaces are constructed.

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.