Ruled threefolds homeomorphic to \({\mathbb{P}}^ 2\times {\mathbb{P}}^ 1\) and their divisors
type: Item
instance of: scholarly article
title: Ruled threefolds homeomorphic to \({\mathbb{P}}^ 2\times {\mathbb{P}}^ 1\) and their divisors
zbMATH Open document ID: 0687.14033
DOI: 10.1007/BF00184155
author: Mirella Manaresi
published in: Geometriae Dedicata
publication date: 1989-01-01T00:00:00Z
review text: Complex projective threefolds \(X=\mathbb{P}(E)\), where E is a topologically trivial rank-2 holomorphic vector bundle on \(\mathbb{P}^ 2\), are studied. Let \(p: X\to\mathbb{P}^ 2\) be the \(\mathbb{P}^ 1\)-bundle projection. The author shows that, except for \(\mathbb{P}^ 2\times \mathbb{P}^1\), p does not admit a holomorphic section, but there exists an irreducible surface \(S\subset X\) such that \(p|_ S:\quad S\to \mathbb{P}^ 2\) is a modification map, which is biholomorphic on \(S\cap p^{-1}(\mathbb{P}^ 2-Z)\), where \(Z\subset \mathbb{P}^2\) is a locally complete intersection of dimension 0 and degree \(d^ 2\), with d any positive integer. Then according to the configuration of Z and some related invariants, she finds conditions for a line bundle on X to have sections, to be ample, to be globally generated and to be very ample. In some instances, when Z consists of simple points, necessary and sufficient conditions for very ampleness are given.
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14J25
zbMATH DE Number: 4125564
zbMATH Keywords: projective threefolds
zbMATH Keywords: line bundle
zbMATH Keywords: very ampleness
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4125564