Remarks on K3 surfaces with non-symplectic automorphisms of order 7

TL;DR

This paper investigates K3 surfaces with non-symplectic automorphisms of orders 7, 21, and 42, establishing uniqueness in certain cases and describing fixed loci for these automorphisms.

Contribution

It demonstrates the uniqueness of K3 surfaces with special fixed loci under order 7 automorphisms and describes fixed loci for automorphisms of orders 21 and 42.

Findings

Uniqueness of K3 surfaces with special fixed loci for order 7 automorphisms

Descriptions of fixed loci for automorphisms of orders 21 and 42

Classification results for non-symplectic automorphisms of these orders

Abstract

In this note, we treat a pair of a K3 surface and a non-symplectic automorphism of order 7m (m=1, 3 and 6) on it. We show that if the fixed locus of a non-symplectic automorphism order 7 is "special" then the pair is unique up to isomorphism. And we describe fixed loci of non-symplectic automorphisms of order 21 and 42.