Bertini's theorem in family
type: Item
instance of: scholarly article
zbMATH Open document ID: 1244.14045
author: Olivier Benoist
publication date: 2012-04-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/0911.1118
full work available at URL: smf bull 139 555-569.php
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 6024215
zbMATH Keywords: Bertini's Theorem
MaRDI profile type: Publication
OpenAlex ID: W2949436213
arXiv ID: 0911.1118
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cites work: Q4143433
cites work: Théoremes de Bertini et applications
cites work: Bertini theorems over finite fields
DOI: 10.24033/BSMF.2619
title: Bertini's theorem in family
published in: Bulletin de la Société Mathématique de France
review text: Let \(K\) be an arbitrary field and consider a (geometrically integral) projective variety \(X\) over \(K\), with \(\dim(X)\geq 2\). Bertini's theorem guarantees that the intersection of \(X\) with a general hypersurface of given degree, is geometrically integral. The author considers the subvarieties \(F^{int}_e\) of the space of hypersurfaces of degree \(e\), which parametrizes hypersurfaces whose intersection with \(X\) is \textit{not} geometrically integral. The target is a lower bound for the codimension of \(F^{int}_e\). The author is indeed able to find sharp lower bounds, which depend on \(e\), \(\dim(X)\) and on the existence of closed points of depth \(1\) in \(X\). As a consequence, the author provides a criterion for the existence of hypersurfaces of degree \(e\), which cut \textit{all} the elements of a given family \(\{X_t\}\) of (geometrically integral) varieties, in geometrically integral divisors.
reviewed by: Luca Chiantini
description: scientific article; zbMATH DE number 6024215