On symplectic and non--symplectic automorphisms of K3 surfaces

TL;DR

This paper studies the conditions under which K3 surfaces with non-symplectic automorphisms also admit symplectic automorphisms of the same order, providing complete classifications for prime orders and exploring automorphisms of order 2p.

Contribution

It offers a complete classification for prime order automorphisms and extends previous results to automorphisms of order 2p on K3 surfaces.

Findings

Complete answer for prime order automorphisms

Examples and partial results for composite orders

Existence of non--symplectic automorphisms of order 2p

Abstract

In this paper we investigate when the generic member of a family of K3 surfaces admitting a non--symplectic automorphism of finite order admits also a symplectic automorphism of the same order. We give a complete answer to this question if the order of the automorphism is a prime number and we provide several examples and partial results otherwise. Moreover we prove that, under certain conditions, a K3 surface admitting a non--symplectic automorphism of prime odd order, $p$, also admits a non--symplectic automorphism of order $2p$. This generalizes a previous result by J. Dillies for $p=3$.