A remark on automorphisms of Kummer surfaces in characteristic p
type: Item
instance of: scholarly article
title: A remark on automorphisms of Kummer surfaces in characteristic p
zbMATH Open document ID: 0612.14040
published in: Journal of Mathematics of Kyoto University
publication date: 1986-01-01T00:00:00Z
review text: The central object of this paper, the Kummer surface S associated with an algebraic curve C of genus \(2,\) was studied in great detail in the last century: If \(\iota\) is the canonical involution in the Jacobian Jac(C) and \(S':=Jac/<1,\iota >\), then S' has \(16\quad ordinary\) double points, and S is the minimal resolution of S'. The present author shows that there exists an elliptic fibration \(\pi: S\to {\mathbb{P}}^ 1_ k\) provided that char(k)\(\neq 2\). Six of the 32 distinguished lines on S provide sections for \(\pi\). If S is considered as an elliptic curve E over the function field K of \({\mathbb{P}}^ 1_ k\) one of these sections serves as unit element of E whereas the other five sections are shown to be K-rational points of infinite order. Here Kodaira's classification of singular fibres of an elliptic fibration plays the key role. This result is used to construct certain compact complex threefolds that are not Kähler.
reviewed by: Frank Herrlich
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 3989545
zbMATH Keywords: non-Kähler threefold
zbMATH Keywords: Kummer surface
zbMATH Keywords: Jacobian
zbMATH Keywords: elliptic fibration
zbMATH Keywords: rational points of infinite order
author: Kenji Ueno
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250520880
OpenAlex ID: W1538126815
DOI: 10.1215/KJM/1250520880
description: scientific article; zbMATH DE number 3989545