Order 3 symplectic automorphisms on \(K3\) surfaces
type: Item
instance of: scholarly article
title: Order 3 symplectic automorphisms on \(K3\) surfaces
zbMATH Open document ID: 1493.14058
author: Alice Garbagnati
author: Yulieth Prieto Montañez
published in: Mathematische Zeitschrift
publication date: 2022-04-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/2102.01207
review text: Ever since the seminal work of [\textit{V. V. Nikulin}, Tr. Mosk. Mat. O.-va 38, 75--137 (1979; Zbl 0433.14024)], there has been interest in symplectic automorphisms of finite order on \(K3\) surfaces. The situation of such automorphiisms of order 2, the so-called Nikulin involutions, is understood quite well, see, e.g., [\textit{B. van Geemen} and \textit{A. Sarti}, Math. Z. 255, No. 4, 731--753 (2007; Zbl 1141.14022)]. In a thorough manner, the article under review progresses to the study of symplectic automorphisms of order 3. A \(K3\) surface together with a symplectic automorphism \(\sigma\) of order \(3\) on \(S\) is constructed in Section 2.1. The two subsequent sections are concerned with the action of \(\sigma\) on the Néron-Severi group \(\mathrm{NS}(S)\) and the lattice \(H^2(S;\mathbb Z)\). This builds the groundwork for the remaining investivations, for, due to Nikulin's results, the action of the automorphism on the \(K3\) lattice -- which is derived in Section 3.1 -- depends only on the order and is otherwise independent from the automorphism (and the surface acted upon). Given a symplectic automorphism \(\sigma\) on a \(K3\) surface \(X\), the minimal resolution \(Y\) of the quotient \(X/\sigma\) is a \(K3\) surface as well and the quotient map lifts to a rational map \(\pi\colon X\dashrightarrow Y\). As the action of the automorphism on the \(K3\) lattice is mostly independent of the automorphism, so is the induced action of this rational map on the \(K3\) lattice, \(\pi_*\), which is provided and intensely studied in Section 3. The families \(\mathcal S\), parametrising \(K3\) surfaces admitting a symplectic automorphism of order \(3\), and \(\mathcal T\), parametrising (the minmal resolutions of) their quotients by said involutions, are described in Section 4, cf. Corollary 4.5 and Corollary 4.10, respectively. The relationship between those families, by means of \(\pi_*\) and its dual \(\pi^*\), is clarified in Section 5. The remaining sections a…
reviewed by: Ben Anthes
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14J50
zbMATH DE Number: 7507815
zbMATH Keywords: \(K3\) surfaces
zbMATH Keywords: symplectic automorphisms
zbMATH Keywords: abelian surfaces
zbMATH Keywords: Shioda-Inose structures
describes a project that uses: Book3264Examples
describes a project that uses: SageMath
MaRDI profile type: Publication
OpenAlex ID: W3217289599
arXiv ID: 2102.01207
cites work: Q4943151
cites work: On certain isogenies between K3 surfaces
cites work: On symplectic quotients of K3 surfaces
cites work: The Coxeter–Todd lattice, the Mitchell group, and related sphere packings
cites work: An Extreme Duodenary Form
cites work: Mirror symmetry for lattice polarized \(K3\) surfaces
cites work: 3264 and all that. A second course in algebraic geometry
cites work: Finite automorphism groups of complex tori of dimension two
cites work: On K3 Surface Quotients of K3 or Abelian Surfaces
cites work: Symplectic automorphisms of prime order on \(K3\) surfaces
cites work: Projective models of K3 surfaces with an even set
cites work: Q4000367
cites work: On K3 surfaces with large Picard number
cites work: Q3901601
cites work: Generalized Shioda-Inose structures on K3 surfaces
cites work: Projective Models of K - 3 Surfaces
cites work: Q4152652
cites work: Nikulin involutions on \(K3\) surfaces
cites work: The two most algebraic K3 surfaces
DOI: 10.1007/S00209-021-02901-9
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description: scientific article