Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces
type: Item
instance of: scholarly article
title: Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces
zbMATH Open document ID: 1246.14057
author: Klaus Hulek
author: Remke Kloosterman
publication date: 2012-02-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/0806.2025
full work available at URL: https://eudml.org/doc/219798
review text: There is an arithmetic part to this paper, and a more purely geometric part. The latter is motivated by the former but is of independent interest. The arithmetic question is to compute the rank of the Mordell-Weil group \(\text{MW}(\pi)\) of an elliptic 3-fold \(\pi: X\to S\) where the base \(S\) is a smooth rational surface, i.e.\ of an elliptic curve over the function field \({\mathbb C}(S)={\mathbb C}(s_1,s_2)\). The method uses the fact that \(X\) has a model \(Y\), given by a minimal Weierstrass equation as a singular hypersurface in a weighted projective space. As long as \(H^4(Y)\) has a pure weight~\(4\) Hodge structure, which is only a very mild condition, the authors prove that \[ \text{rank}\text{MW}(\pi)=\text{rank}\big(H^{2,2}(H^4(Y{\mathbb C}))\cap H^4(Y,{\mathbb Z})\big)-1. \] If \(X\) is general then \(Y\) is quasismooth and in that case \(\text{MW}(\pi)=0\). The authors give explicit conditions on the discriminant under which this happens, and explain why the problem is harder for elliptic surfaces The second part of the paper aims to compute the cohomology of a hypersurface in weighted projective space, under conditions on the singularities that \(Y\) might often satisfy. The authors extend the methods used by Dimca for hypersurfaces with isolated semi-weighted homogeneous singularities, so as to compute \(H^4(Y)\) provided that the codimension~\(2\) singularities are of transverse Du Val type and the ccodimension \(3\) singularities are contact equivalent to weighted homogeneous singularities: this is quite hard. They are able to verify that these conditions are satisfied by the models of many elliptic threefolds, and thus to compute their Mordell-Weil ranks.
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 32S20
Mathematics Subject Classification ID: 32S35
Mathematics Subject Classification ID: 32S50
zbMATH DE Number: 6002992
zbMATH Keywords: Mordell-Weil group of elliptic threefolds
zbMATH Keywords: cohomology of singular varieties
zbMATH Keywords: mixed Hodge stuctures
reviewed by: G. K. Sankaran
MaRDI profile type: Publication
OpenAlex ID: W2962817563
arXiv ID: 0806.2025
cites work: Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
cites work: Double solids
cites work: The Noether-Lefschetz Locus of Regular Elliptic Surfaces with Section and p g ≥2
cites work: Defect of a nodal hypersurface
cites work: Filtrations de Hodge et par l'ordre du pôle pour les hypersurfaces singulières
cites work: Q3764252
cites work: Betti numbers of hypersurfaces and defects of linear systems
cites work: Singularities and topology of hypersurfaces
cites work: Q3474764
cites work: On the periods of certain rational integrals. I, II
cites work: Q4507537
cites work: Elliptic K3 Surfaces with Geometric Mordell–Weil Rank 15
cites work: Higher Noether-Lefschetz loci of elliptic surfaces
cites work: A different method to calculate the rank of an elliptic threefold
cites work: Q3708118
cites work: Q4000367
cites work: Q3982645
cites work: Mixed Hodge Structures
cites work: Algebraic cycles on certain desingularized nodal hypersurfaces
cites work: Q4116546
cites work: Q5488920
cites work: Q4420474
cites work: Arithmetic on elliptic threefolds
cites work: Q3804827
DOI: 10.5802/AIF.2637
published in: Annales de l’institut Fourier
description: scientific article