On certain Mordell-Weil lattices of hyperelliptic type on rational surfaces
type: Item
instance of: scholarly article
title: On certain Mordell-Weil lattices of hyperelliptic type on rational surfaces
zbMATH Open document ID: 0984.14013
DOI: 10.1007/BF02984108
author: Nguyen Khac Viet
published in: Journal of Mathematical Sciences (New York)
publication date: 2001-01-16T00:00:00Z
review text: The main object of the paper under review is the Mordell-Weil lattice (MWL) of a surface \(X\) admitting a (relatively minimal) fibration \(f\colon X\to C\) into curves of genus \(g\geq 1\) over an algebraic curve \(C\) defined over an algebraically closed field \(k\). By definition [see, for example, \textit{T.~Shioda}, Proc. Japan Acad., Ser. A 68, 247-250 (1992; Zbl 0788.14021)], MWL is the group \(J(K)\) (modulo torsion) viewed together with the height pairing (here \(K=k(C)\) and \(J\) stands for the Jacobian of the generic fibre of \(f\)). In the case where \(X\) is a rational elliptic surface, MWL turned out to be closely related to the root lattice \(E_8\). The author considers a natural generalization of this phenomenon. Namely, for \(m\geq 2\) let \[ \Gamma _{4m}=\left\{\sum_{i=1}^{4m}x_ie_i : x_i\in 2\mathbb Z,\;x_1\equiv\dots\equiv x_{4m}\pmod{\mathbb Z},\;\sum_{i=1}^{4m}x_i\equiv 0\pmod{2\mathbb Z}\right\}. \] This is a unimodular lattice in \(\mathbb R^{4m}\) which can be viewed as a natural generalization of the root lattice \(E_8\). (Another common notation for \(\Gamma _{4m}\) is \(D_{4m}^+\), the unique lattice contained strictly between \(D_{4m}\) and its dual.) Such lattices appeared as the MWL for higher genus fibrations on a rational surface over \(\mathbb C\) [see \textit{M.-H.~Saito} and \textit{K.-I.~Sakakibara}, J. Math. Kyoto Univ. 34, 859-871 (1994; Zbl 0860.14034) and \textit{K.-V.~Nguyen}, Osaka J. Math. 34, 101-114 (1997; Zbl 0891.14008)]. The first main result of the paper under review is a generalization of this observation to the case of an arbitrary ground field \(k\) of characteristic \(\neq 2\); in particular, it says that for \(g\geq 2\) a surface \(X\) with maximal possible Mordell-Weil rank is a rational surface admitting a hyperelliptic fibration (in other words, \(X\) can be obtained as a blow-up of a linear pencil of hyperelliptic curves on a Hirzebruch surface \(\Sigma _e\) with \(0\leq e\leq g\)), and its MWL is isometric…
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14M20
zbMATH DE Number: 1551884
zbMATH Keywords: rational surface
zbMATH Keywords: fibration
zbMATH Keywords: Mordell-Weil lattice
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
cites work: Q5580319
cites work: Q4039784
cites work: Q5813959
cites work: BETTI AND PICARD NUMBERS OF ABSTRACT ALGEBRAIC SURFACES
cites work: Q4043127
cites work: Bogomolov conjecture over function fields for stable curves with only irreducible fibers (Version 2.0)
cites work: On Beauville's conjecture and related topics
cites work: On upper bounds of virtual Mordell-Weil ranks
cites work: Q4195021
cites work: Q3982645
cites work: Q5571531
cites work: On Mordell-Weil lattices of higher genus fibrations on rational surfaces
cites work: Q5203144
cites work: Mordell-Weil lattices for higher genus fibration
cites work: Q4263537
cites work: Pencils on an Algebraic Variety and a New Proof of a Theorem of Bertini
full work available at URL: https://doi.org/10.1007/bf02984108
OpenAlex ID: W2068327486
Recommended article: On Mordell-Weil lattices of bielliptic fibrations on rational surfaces
Recommended article: On the Mordell-Weil lattices of nonhyperelliptic type
Recommended article: On Mordell-Weil lattices of higher genus fibrations on rational surfaces
Recommended article: On Mordell-Weil lattices for non-hyperelliptic fibrations on surfaces with zero geometric genus and irregularity
Recommended article: Q5203144
Recommended article: Fibred rational surfaces with extremal Mordell-Weil lattices
Recommended article: Existence of a rational elliptic surface with a given Mordell-Weil lattice
Recommended article: Mordell-Weil lattice of higher genus fibration on a Fermat surface
Recommended article: Mordell-Weil lattices and toric decompositions of plane curves
Recommended article: Q4892941
description: scientific article; zbMATH DE number 1551884