Torsion sections of elliptic surfaces
type: Item
instance of: scholarly article
title: Torsion sections of elliptic surfaces
zbMATH Open document ID: 0843.14015
DOI: 10.1007/BF02559607
author: Rick Miranda
author: Peter F. Stiller
published in: Arkiv för Matematik
publication date: 1996-08-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/alg-geom/9310006
review text: A semistable elliptic surface is a fibration \(p : X \to C\) of a smooth compact surface \(X\) onto a smooth curve \(C\) whose general fibre is elliptic and with semistable singular fibres (here \(p\) is assumed to have a section). The zero section \(S_0\) meets only a component \(C^0_J\) of a singular fibre \(F^J\). The authors consider a torsion section \(S\) of order \(p\) and those fibres \(F^J\) such that their component hit by \(S\) is still \(C^0_J\). They build up a number taking into account these components and, among them, those hit in a point of \(C^0_J\) having a ``fixed'' coordinate in a previously given suitable coordinate choice in \(C^0_J\). In the main result the authors prove that this number is \(2/p - 1\) and so independent of \(S\) and of the coordinate ``fixed'' among the possible ones. They give a proof based on computation on elliptic modular surfaces and another one via an explicit computation.
Mathematics Subject Classification ID: 14J27
zbMATH DE Number: 792857
zbMATH Keywords: semistable elliptic surface
zbMATH Keywords: singular fibres
zbMATH Keywords: torsion section
reviewed by: Marino Palleschi
MaRDI profile type: Publication
arXiv ID: alg-geom/9310006
cites work: Q4099744
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OpenAlex ID: W2101690206
description: scientific article