Non-symplectic automorphisms of \(K3\) surfaces with one-dimensional moduli space
In this paper the authors consider one-dimensional moduli spaces of \(K3\) surfaces carrying a purely non-symplectic automorphism of order \(n\). In particular they prove that, as expected, the moduli space is one dimensional if \(n= 11, 15, 16, 20, 22, 24, 30\). For all these values of \(n\), they classify pairs \((X,\sigma)\) such that \(\dim(V^{\sigma})=2\), (where \(V^{\sigma}\) is the \(\xi_{n}\)-eigenspace of \(\sigma^{*}\) in \(H^{2}(X,\mathbb{C})\)) they determine the dimensions of the eigenspaces of \(\sigma^{*}\) in \(H^{2}(X,\mathbb{C})\) and compute the Néron--Severi lattice of a very general pair. The orders \(n=11\) and \(n=16\) had been previously studied in [\textit{M. Artebani} et al., Math. Z. 268, No. 1--2, 507--533 (2011; Zbl 1218.14024); \textit{K. Oguiso} and \textit{D.-Q. Zhang}, Pure Appl. Math. Q. 7, No. 4, 1657--1673 (2011; Zbl 1316.14070); \textit{D. Al Tabbaa} et al., J. Korean Math. Soc. 53, No. 6, 1237--1260 (2016; Zbl 1352.14025)]. Theorem 1.1 collects these results, more precisely it provides a list (Table 1) of Néron-Severi lattices for a very general \(K3\) surface \(X\) with a purely non-symplectic automorphism \(\sigma\) of order \(n \geq 2 \) such that \(\varphi(n)=8\) or \(10\) and \(\dim(V^{\sigma})=2\). Using the classification of Theorem 1.1 the authors provide explicit birational models for a very general pair \((X, \sigma)\) (up to admit some generality assumptions). More precisely in Theorem 1.2 they prove that the pair \((X, \sigma)\) belongs to the one of the one-dimensional families of \(K3\) surfaces with non-symplectic automorphisms described in Table 2. Finally in Corollary 1.3 the one dimensional components of the moduli space of \(K3\) surfaces with a purely non-symplectic automorphism of order \(n\) are counted: there is a unique one-dimensional component for \(n=20, 22, 24\), there are three components for \(n=15\) and two components for \(n=11,16,30\).
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