On the extendability of elliptic surfaces of rank two and higher
type: Item
instance of: scholarly article
title: On the extendability of elliptic surfaces of rank two and higher
zbMATH Open document ID: 1167.14036
author: Angelo Felice Lopez
author: José Carlos Sierra
author: Roberto Muñoz
publication date: 2009-04-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/0806.1440
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10394
review text: A natural approach to study projective algebraic variety is the so called Apollonius method. That is consider an hyperplane section and then reconstruct the geometry of the ambient variety via the one of its section. It is therefore quite natural to understand which varieties can be embedded as hyperplane sections of other varieties. Besides many classical results, a fundamental work of \textit{A. J. Sommese} [Math. Ann. 221, 55--72 (1976; Zbl 0306.14006)] gave many criteria for extendability and non extendability of algebraic varieties. This area is called adjunction theory [\textit{M. C. Beltrametti, J. J. Sommese}, The adjunction theory of complex projective varieties. De Gruyter Expositions in Mathematics. 16. (Berlin): de Gruyter. (1995; Zbl 0845.14003)], and attracted many attention in the last decades, also for its relations with Minimal Model Program. The paper under review considers this problem for smooth surfaces that admits an elliptic fibrations. The main result is a list of the possible embeddings of these surfaces, of Picard rank 2, as hyperplane sections of mildly singular 3-folds. This is done via a subtle study of the geometry of the surface in relation with its possible three dimensional extensions. The result is then improved with stronger hypothesis on the surface and many interesting examples are given.
reviewed by: Massimiliano Mella
Mathematics Subject Classification ID: 14N30
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14J30
zbMATH DE Number: 5541203
zbMATH Keywords: elliptic surfaces
zbMATH Keywords: adjunction theory
zbMATH Keywords: hyperplane sections
MaRDI profile type: Publication
OpenAlex ID: W2963717437
arXiv ID: 0806.1440
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DOI: 10.5802/AIF.2432
published in: Annales de l’institut Fourier
description: scientific article