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

The moduli space of K3 surfaces $X$ with a purely non-symplectic automorphism $σ$ of order $n\geq 2$ is one dimensional exactly when $φ(n)=8$ or $10$. In this paper we classify and give explicit equations for the very general members $(X,σ)$ 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.

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