Bertini theorems and Lefschetz pencils over discrete valuation rings, with applications to higher class field theory
type: Item
instance of: scholarly article
title: Bertini theorems and Lefschetz pencils over discrete valuation rings, with applications to higher class field theory
zbMATH Open document ID: 1267.14010
DOI: 10.1090/S1056-3911-2012-00570-0
author: Uwe Jannsen
author: Shuji Saito
published in: Journal of Algebraic Geometry
publication date: 2012-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/0911.1470
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14C21
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 6086796
zbMATH Keywords: hyperplane section
zbMATH Keywords: Lefschetz pencil
zbMATH Keywords: Bertini theorem
zbMATH Keywords: discrete valuation ring
Wikidata QID: Q102060881
MaRDI profile type: Publication
arXiv ID: 0911.1470
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review text: Let \(X\subset {\mathbb P} ^{N}_{L}\) be a smooth quasi-projective variety over a field \(L\). A subscheme \(Z\subset {\mathbb P} ^{N}_{L}\) intersects \(X\) transversally if \(X\cdot Z=X{\times}_{{\mathbb P}^{N}}Z\) is smooth and of pure codimension \(\mathrm{codim}_{{\mathbb P}^{N}}(Z)\) in \(X.\) The Bertini theorem asserts that for infinite \(L\), there exists an \(L\)-rational hyperplane that intersects \(X\) transversally. Good families of hyperplane sections - Lefschetz pencils - proved to be useful in algebraic geometry. However, for arithmetic questions it is natural to consider models over Dedekind domains or discrete valuation rings (for local questions). The authors extend the classical constructions mentioned above to the case of schemes over a discrete valuation ring \(A.\) They show the existence of good hyperplane sections for schemes over \(A\) with good or (quasi-) semi- stable reduction and the existence of (good) Lefschetz pencils for schemes with good or ordinary quadratic reduction. As an applicalion the authors prove a nice result concerning the reciprocity map introduced for smooth projective varieties by Bloch, Kato and Saito.
reviewed by: Piotr Krasoń
description: scientific article; zbMATH DE number 6086796