Non-symplectic automorphisms of K3 surfaces with one-dimensional moduli space

TL;DR

This paper classifies K3 surfaces with one-dimensional moduli spaces arising from purely non-symplectic automorphisms of specific orders, providing explicit equations and describing the structure of these moduli components.

Contribution

It offers a complete classification and explicit equations for the maximal dimension moduli spaces of K3 surfaces with such automorphisms, detailing the number of components for each order.

Findings

Unique one-dimensional component for n=20,22,24

Three components for n=15

Two components for other cases

Abstract

The moduli space of K3 surfaces $X$ with a purely non-symplectic automorphism $\sigma$ of order $n\geq 2$ is one dimensional exactly when $\varphi(n)=8$ or $10$. In this paper we classify and give explicit equations for the very general members $(X,\sigma)$ of the irreducible components of maximal dimension of such moduli spaces. In particular we show that there is a unique one-dimensional component for $n=20,22, 24$, three irreducible components for $n=15$ and two components in the remaining cases.