A special configuration of 12 conics and generalized Kummer surfaces
type: Item
instance of: scholarly article
title: A special configuration of 12 conics and generalized Kummer surfaces
zbMATH Open document ID: 1503.14035
author: Xavier Roulleau
author: Alessandra Sarti
author: David Kohel
published in: Manuscripta Mathematica
publication date: 2022-10-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/2004.11421
review text: In analogy with classical Kummer surfaces, a \textit{generalized Kummer surface} is the minimal desingularization of the quotient of an abelian surface \(A\) by a symplectic group \(G\subseteq \mathrm{Aut}(A)\). In this article the authors focus on the case \(G\cong \mathbb{Z}/3\mathbb{Z}\): the minimal desingularization \(\mathrm{Kum}_3(A)\) of a quotient \(A/G\) is a \(K3\) surface with nine disjoint \(A_2\)-configurations (i.e. pairs of smooth rational curves \(C,C'\) with \(C\cdot C'=1\)), arising over the nine cusps of \(A/G\). We call this a \textit{\(9A_2\)-configuration}. Conversely, any \(9A_2\) configuration on a \(K3\) surface \(X\) leads to a generalized Kummer structure on \(X\), i.e. an abelian surface \(A\) such that \(X\cong \mathrm{Kum}_3(A)\). The goal of the article under review is to investigate the geometry of a particular family of generalized Kummer surfaces, previously studied by \textit{C. Birkenhake} and \textit{H. Lange} [Manuscr. Math. 85, No. 3--4, 393--407 (1994; Zbl 0821.14027)]. For generic \(\lambda\), the dual of the cubic curve \(E_\lambda: x^3+y^3+z^3-3\lambda xyz=0\) is a sextic curve \(C_\lambda\) with nine \(A_2\)-singularities. Thus the double cover \(X_\lambda\) of \(\mathbb{P}^2\), branched along \(C_\lambda\) is a \(K3\) surface with a generalized Kummer structure. The authors show that there are precisely \(12\) conics in \(\mathbb{P}^2\) passing through \(6\) of the \(9\) cusps of \(C_\lambda\). Their pullbacks on \(X_\lambda\) are \(24\) smooth rational curves, and the authors use these curves to produce nine new \(9A_2\)-configurations on \(X_\lambda\). Moreover they prove that these new \(9A_2\)-configurations are in the \(\mathrm{Aut}(X_\lambda)\)-orbit of the original \(9A_2\)-configuration. Finally, the authors realize the surfaces \(X_\lambda\) as degree \(8\) non-complete intersections in \(\mathbb{P}^5\), and they obtain a Weierstrass model for them.
reviewed by: Giacomo Mezzedimi
Mathematics Subject Classification ID: 14J28
zbMATH DE Number: 7602949
zbMATH Keywords: generalized Kummer
zbMATH Keywords: \(K3\) surface
describes a project that uses: Magma
MaRDI profile type: Publication
OpenAlex ID: W3195389853
arXiv ID: 2004.11421
cites work: The Hesse pencil of plane cubic curves
cites work: Q4889534
cites work: Réseaux de Kummer et surfaces K3. (Kummer lattices and K3 surfaces)
cites work: \(K3\) surfaces with nine cusps
cites work: Q4949380
cites work: Q4434669
cites work: A family of abelian surfaces and curves of genus four
cites work: The Magma algebra system. I: The user language
cites work: Q4023809
cites work: Oka's conjecture on irreducible plane sextics
cites work: Q4650879
cites work: Chilean configuration of conics, lines and points
cites work: Q4195061
cites work: Minimal Siegel modular threefolds
cites work: Lectures on K3 Surfaces
cites work: Q4000367
cites work: On K3 surfaces with large Picard number
cites work: Q4195021
cites work: On the geometry of K3 surfaces with finite automorphism group and no elliptic fibrations
cites work: Construction of Nikulin configurations on some Kummer surfaces and applications
cites work: Explicit Nikulin configurations on Kummer surfaces
cites work: Transcendental lattices of some K 3-surfaces
cites work: Projective Models of K - 3 Surfaces
cites work: Q5194786
cites work: Q4317891
cites work: On elliptic \(K3\) surfaces.
cites work: On elliptic modular surfaces
cites work: Q4192187
DOI: 10.1007/S00229-021-01334-2
description: scientific article