The number of reducible hypersurfaces in a pencil. -- Erratum
type: Item
instance of: scholarly article
title: The number of reducible hypersurfaces in a pencil. -- Erratum
zbMATH Open document ID: 0799.14024
DOI: 10.1007/BF01232434
author: Angelo Vistoli
published in: Inventiones Mathematicae
publication date: 1993-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/144103
review text: Let \(A\) be a pencil of hypersurfaces on a smooth projective variety \(X\) over an algebraically closed field of characteristic 0, whose general element is irreducible. The total order of reducibility \(r(A)\) of \(A\) is defined as the number of reducible elements of \(A\), counted with multiplicity. Let \(n(A)\) be the rank of the subgroup of the Néron- Severi group of \(X\) generated by all the components of all the elements of \(A\). In this note the difference \(r(A) - n(A)\) is expressed in terms of the base number, another invariant of \(A\). This character can be very difficult to compute; however some bounds are provided. Some applications are also supplied including a geometrical characterization of minimal rational surfaces. The erratum is warning that some pages were wrongly paginated.
reviewed by: Marino Palleschi
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 14N10
zbMATH DE Number: 4217789
zbMATH Keywords: pencil of hypersurfaces
zbMATH Keywords: order of reducibility
zbMATH Keywords: Néron-Severi group
zbMATH Keywords: base number
MaRDI profile type: Publication
cites work: Q5749404
cites work: Q4039844
cites work: Reducibility of polynomials in two variables
cites work: Profinite Groups, Arithmetic, and Geometry. (AM-67)
cites work: The total reducibility order of a polynomial in two variables
description: scientific article; zbMATH DE number 4217789