Supersingular \(K3\) surfaces in odd characteristic and sextic double planes
type: Item
instance of: scholarly article
title: Supersingular \(K3\) surfaces in odd characteristic and sextic double planes
zbMATH Open document ID: 1060.14058
author: Ichiro Shimada
published in: Mathematische Annalen
publication date: 2004-07-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0309451
review text: Let \(X\) be a \(K3\) surface defined over an algebraically closed field of positive characteristic \(p\neq 2\). According to Shioda, \(X\) is said to be supersingular if its Picard lattice has rank 22. The main result of this paper is the following: Theorem. If \(X\) is a supersingular \(K3\) surface, then there exists a separable degree 2 morphism \(f: X\to{\mathbb P}^2\). (In other words, \(X\) is a so-called sextic double plane.) The idea of the proof is to use lattice theoretic arguments to show the existence of a nef class \(L\in \text{Pic}(X)\) with \(L^2=2\). Then the morphism \(f\) is just the map given by the linear system \(| L| \). The author also describes the branch locus of \(f\), which is a plane sextic with at most simple singularities. Hence, if \(X\to Y\to{\mathbb P}^2\) is the Stein factorization of \(f\), then the surface \(Y\) is birational to \(X\) and it has at most canonical singularities. The characteristic 2 case had already been treated by the author in an earlier paper [\textit{I. Shimada}, Math. Comput. 73, No. 248, 1989--2017 (2004; Zbl 1052.14041)], in which he had shown the existence of an inseparable degree 2 map \(f\colon X\to {\mathbb P}^2\).
reviewed by: Rita Pardini
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14Q10
Mathematics Subject Classification ID: 11H55
zbMATH DE Number: 2078025
zbMATH Keywords: Picard lattice
MaRDI profile type: Publication
OpenAlex ID: W2033265212
arXiv ID: math/0309451
DOI: 10.1007/S00208-003-0494-X
Recommended article: On Kummer type construction of supersingular \(K3\) surfaces in characteristic 2
Recommended article: Supersingular \(K3\) surfaces for large primes. With an Appendix by Andrew Snowden.
Recommended article: Kummer surfaces for the self-product of the cuspidal rational curve
Recommended article: On normal \(K3\) surfaces
Recommended article: Q3728125
Recommended article: Rational double points on supersingular 𝐾3 surfaces
Recommended article: Kummer surfaces associated with group schemes
Recommended article: Unirationality of certain supersingular \(K3\) surfaces in characteristic 5
Recommended article: Supersingular \(K3\) surfaces are unirational
Recommended article: MODULI CURVES OF SUPERSINGULAR K3 SURFACES IN CHARACTERISTIC 2 WITH ARTIN INVARIANT 2
description: scientific article