Threefolds whose hyperplane sections are elliptic surfaces
type: Item
instance of: scholarly article
title: Threefolds whose hyperplane sections are elliptic surfaces
zbMATH Open document ID: 0626.14031
author: Harry D'Souza
published in: Pacific Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: We classify pairs \((X,S)\) where \(X\) is a smooth complex projective threefold and \(S\) is a smooth ample divisor in \(X\). Moreover S is elliptic and \(\kappa (S)=1\).We use the logarithmic Kodaira dimension of \((X,S)\) as the basis of classification. Sommese studied such pairs, and he has shown that such pairs \((X,S)\) can be reduced to \((X',S')\), where \(S'\) is ample in \(X'\), and \(S'\) is a minimal model of \(S\). In the case when \(S\) is elliptic, with \(h^{1,0}(S)\neq 0\) he showed that one obtains a surjective morphism p, from X onto a smooth curve \(Y\) such that this morphism restricted to \(S\), is a reduced elliptic fibration. Shepherd- Barron proved the same result, using Mori's methods without the restriction on \(h^{1,0}(S)\). We state these results in section 0. We show that the general fibres of p are del Pezzo surfaces and classify these in the case where they are of degrees 1, 2, 3, 4, 7, 8 and 9. We also show in the degree 9 case, it is indeed a \({\mathbb{P}}^ 2\)-bundle over Y. In the degree 8 (\(\cong {\mathbb{P}}^ 1\times {\mathbb{P}}^ 1)\) case we have a birational morphism to a \({\mathbb{P}}^ 2\)-bundle.
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 4017085
zbMATH Keywords: threefold
zbMATH Keywords: smooth ample divisor
zbMATH Keywords: logarithmic Kodaira dimension
zbMATH Keywords: elliptic fibration
MaRDI profile type: Publication
full work available at URL: pjm.1988.134.57
OpenAlex ID: W2100718251
DOI: 10.2140/PJM.1988.134.57
Recommended article: On algebraic threefolds whose hyperplane sections are Enriques surfaces
Recommended article: Bipolarized threefolds with hyperelliptic curve sections.
Recommended article: Elliptic fibrations of a \(K3\) surface
Recommended article: Elliptic 3-folds and non-Kähler 3-folds
Recommended article: The classification of hyperelliptic threefolds
Recommended article: Q3752516
Recommended article: Elliptic 𝐾3 surfaces with 𝐩^{𝐧}-torsion sections
Recommended article: Elliptic Fibrations on K3 Surfaces
Recommended article: Q3378916
Recommended article: Elliptic fibrations of some extremal \(K3\) surfaces
description: scientific article; zbMATH DE number 4017085