Persson's list of singular fibers for a rational elliptic surface
type: Item
instance of: scholarly article
title: Persson's list of singular fibers for a rational elliptic surface
zbMATH Open document ID: 0722.14022
DOI: 10.1007/BF02571235
author: Rick Miranda
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174168
review text: This paper classifies all configurations of singular fibers of rational elliptic surfaces (i.e. elliptic fibrations over \({\mathbb{P}}^ 1)\). This classification is simultaneously given by \textit{U. Persson} [ibid. 205, No.1, 1-47 (1990; see the preceding review)] by other methods [see also Persson's review of his joint paper with the author, Math. Z. 193, 537- 558 (1986; Zbl 0652.14003 for a general perspective of the problem]. A first list of possibilities is given by using the same numerical restrictions as in the previous papers. The author's key tool for both ruling out the impossible configurations and proving the existence of the others is the detailed study of the J-map. This map, which associates to each point of the base \({\mathbb{P}}^ 1\) the j-invariant of its fiber, has the property that its ramification at points 0, 1 and \(\infty\) is closely related to the configuration of singular fibers. The author's list contains 279 configurations, which contrasts with the 289 cases found by Persson; the reason is that the Mordell-Weil group of sections of 10 of them can either be torsion-free or have a 2-torsion element, so that they appear twice in Persson's list.
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14J17
zbMATH DE Number: 4189896
zbMATH Keywords: Kodaira singular fiber
zbMATH Keywords: configurations of singular fibers of rational elliptic surfaces
reviewed by: Enrique Arrondo
MaRDI profile type: Publication
cites work: Q5749404
cites work: On the semi-universal deformation of a simple-elliptic hypersurface singularity part II: The discriminant
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cites work: INTEGRAL SYMMETRIC BILINEAR FORMS AND SOME OF THEIR APPLICATIONS
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cites work: Q3661722
OpenAlex ID: W2010448188
description: scientific article; zbMATH DE number 4189896