The minimal height of Jacobian fibrations on \(K3\) surfaces
type: Item
instance of: scholarly article
title: The minimal height of Jacobian fibrations on \(K3\) surfaces
zbMATH Open document ID: 0893.14012
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1997-09-22T00:00:00Z
review text: A Jacobian fibration \(\Phi:X\to C\) on a surface \(X\) is an elliptic fibration on \(X\) with a global section. Its Mordell-Weil group \(\Lambda\) (automorphisms which act as translations on fibers) is equipped with a bilinear form \(\phi\), called ``height pairing''. The minimum for the height is known for rational surfaces, since the Jacobian fibrations on them are classified. The author proves here that on a K3 surface, the minimum for the height is \(11/420\) and this value is attained by some Jacobian fibrations. The proof is obtained by studying, with the aid of a computer, all the possibilities for the root lattice of \((\Lambda,\phi)\).
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14D99
zbMATH DE Number: 1044137
zbMATH Keywords: Jacobian fibration
zbMATH Keywords: elliptic fibration
zbMATH Keywords: Mordell-Weil group
zbMATH Keywords: height pairing
zbMATH Keywords: K3 surface
author: Ken'ichi Nishiyama
reviewed by: Luca Chiantini
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225295
OpenAlex ID: W1966569422
cites work: Q5749404
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cites work: A TORELLI THEOREM FOR ALGEBRAIC SURFACES OF TYPEK3
cites work: Q5203144
cites work: Existence of a rational elliptic surface with a given Mordell-Weil lattice
cites work: Q4152652
DOI: 10.2748/TMJ/1178225295
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description: scientific article; zbMATH DE number 1044137