Classification of order sixteen non-symplectic automorphisms on \(K3\) surfaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1352.14025
author: Dima Al Tabbaa
author: Alessandra Sarti
author: Shingo Taki
published in: Journal of the Korean Mathematical Society
publication date: 2016-12-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/1409.5803
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14J50
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 6658982
zbMATH Keywords: non-symplectic automorphisms
zbMATH Keywords: \(K3\) surfaces
zbMATH Keywords: Picard lattices
MaRDI profile type: Publication
arXiv ID: 1409.5803
DOI: 10.4134/JKMS.J150339
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title: Classification of order sixteen non-symplectic automorphisms on \(K3\) surfaces
review text: This paper classifies complex \(K3\) surfaces with non-symplectic automorphisms of order \(16\) in full generality.NEWLINENEWLINELet \(X\) be a \(K3\) surface, \(\omega_X\) a generator of \(H^{2,0}(X)\), \(\sigma\) an order \(16\) automorphism such that \(\sigma^*\omega_X=\zeta_{16}\omega_X\), where \(\zeta_{16}\) denotes a primitive \(16\)-th root of unity. It is assumed throughout the paper that \(\sigma^8\) acts as the identity on \(\mathrm{Pic}(X)\). Under this situation, the rank of \(\mathrm{Pic}(X)\) can only be \(6\) or \(14\).NEWLINENEWLINEThe first results are about the fixed locus of \(\sigma\) and that of \(\sigma^4\): if \(\mathrm{Fix}(\sigma)\) contains a curve, its genus is zero, and \(\mathrm{Fix}(\sigma^4)\) always contains at least a curve of genus \(0\) or \(1\).NEWLINENEWLINEIf \(\mathrm{rk}\, \mathrm{Pic}(X)=6\), there are two possibilities: the lattice structure of \(\mathrm{Pic}(X)\), the number \(N\) of \(\sigma\)-fixed points, and the number \(k\) of \(\sigma\)-fixed rational curves are classified as NEWLINE\[NEWLINE(\mathrm{Pic}(X),N,k)=(U\oplus D_4, 6,1),\quad\text{{or}}\quad (U(2)\oplus D_4, 4,0).NEWLINE\]NEWLINENEWLINENEWLINEIf \(\mathrm{rk}\, \mathrm{Pic}(X)=14\), there are five possibilities: the number \(N\) of isolated \(\sigma\)-fixed points and the number \(k\) of \(\sigma\)-fixed rational curves are \((N,k)=(8,1)\) or \((6,0)\), and if \(\mathrm{Fix}(\sigma^4)\) contains at least a curve and its genus is at most \(0\), there are three cases for \((\mathrm{Pic}(X),N,k)\): NEWLINE\[NEWLINE(\mathrm{Pic}(X),N,k)=(U\oplus D_4\oplus E_8, 12,1),(U(2) \oplus D_4\oplus E_8, 4,0), (U(2)\oplus D_4\oplus E_8,10,1).NEWLINE\]NEWLINENEWLINENEWLINEThese classification results imply the non-existence of a \(K3\) surface \(X\) with \mathrm{Pic}ard number \(14\) with a non-symplectic automorphism of order \(16\) acting trivially on \(\mathrm{Pic}(X)\). Also it is shown that if the action of the automorphism of order \(16\) is trivial on \(\mathrm{Pi…
reviewed by: Noriko Yui
description: scientific article; zbMATH DE number 6658982