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

Abstract

The moduli space of K3 surfaces $X$ with a purely non-symplectic automorphism $\sigma$ of order $n\ge 2$ is one dimensional exactly when $\phi (n)=8$ 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.