Ruled fibrations on normal surfaces
type: Item
instance of: scholarly article
title: Ruled fibrations on normal surfaces
zbMATH Open document ID: 0655.14014
published in: Journal of the Mathematical Society of Japan
publication date: 1988-01-01T00:00:00Z
review text: Let Y be a complex normal projective surface. A ruled fibration on Y is a surjective morphism \(p: Y\to B\) over a smooth curve, whose general fibre is isomorphic to \({\mathbb{P}}^ 1.\) A ruled fibration is called minimal if its fibres contain no exceptional curve C of the first kind (i.e. an irreducible curve C with \(K_ Y\cdot C<0\) and \(C^ 2<0\), where \(K_ Y\) stands for a canonical divisor). The structure of normal surfaces admitting a minimal ruled fibration p is studied. The author shows that all singular fibres of p are multiple fibres containing one or two points of Sing(Y). In particular this allows him to correct a statement in one of his previous papers [J. Reine Angew. Math. 366, 121-128 (1986; Zbl 0582.14011)]. By using a suitable invariant related to the amount of Sing(Y), the author extends to the normal case some results, which are known for smooth ruled surfaces, e.g. the ampleness criterion and the Nagata inequality for the self-intersection numbers of all sections of p. Moreover the classification of Y in terms of its anti-Kodaira dimension and the numerical type of \(-K_ Y\) are given. As a last thing the author proves that \(-K_ Y\) is ample if and only if Y either admits another minimal ruled fibration or contains an exceptional curve of the first kind.
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 4069068
zbMATH Keywords: Weil divisor
zbMATH Keywords: structure of normal surfaces admitting a minimal ruled fibration
zbMATH Keywords: ampleness criterion
zbMATH Keywords: Nagata inequality for the self- intersection numbers
zbMATH Keywords: classification
author: Fumio Sakai
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04020249
OpenAlex ID: W2049180812
DOI: 10.2969/JMSJ/04020249
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description: scientific article; zbMATH DE number 4069068