The arithmetic of certain del Pezzo surfaces and \(K3\) surfaces
type: Item
instance of: scholarly article
title: The arithmetic of certain del Pezzo surfaces and \(K3\) surfaces
zbMATH Open document ID: 1268.14020
author: Nguyen Ngoc Dong Quan
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2012-10-26T00:00:00Z
review text: It is a classical result of Hasse that smooth quartic hypersurfaces always satisfy the Hasse principle. A degree 4 del Pezzo surface is the smooth intersection of two quadratic hypersurfaces of dimension 4. The author constructs degree 4 del Pezzo surfaces for which the Hasse principle fails because of the Brauer-Manin obstruction. They give explicit equations for the two quadric hypersurfaces defining the surface. One of the surfaces provided has been studied by \textit{B. J. Birch} and \textit{H. P. F. Swinnerton-Dyer} [J. Reine Angew. Math. 274--275, 164--174 (1975; Zbl 0326.14007)]. The author then uses the degree 4 del Pezzo surfaces to construct a family of \(K3\) surfaces violating the Hasse principle also. These surfaces are double covers of the del Pezzo branched over a smooth conic.
reviewed by: Amanda Knecht
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 11G35
zbMATH DE Number: 6099153
zbMATH Keywords: Hasse principle
zbMATH Keywords: degree 4 del Pezzo surfaces
zbMATH Keywords: \(K3\) surfaces
zbMATH Keywords: Brauer-Manin obstruction
MaRDI profile type: Publication
full work available at URL: jtnb.805
OpenAlex ID: W2032704789
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DOI: 10.5802/JTNB.805
description: scientific article; zbMATH DE number 6099153