Fano manifolds and quadric bundles
type: Item
instance of: scholarly article
title: Fano manifolds and quadric bundles
zbMATH Open document ID: 0817.14020
DOI: 10.1007/BF02572403
author: Jarosław A. Wiśniewski
published in: Mathematische Zeitschrift
publication date: 1995-07-27T00:00:00Z
full work available at URL: https://eudml.org/doc/174567
review text: Here the author proves (using Mori theory, the deformations of rational curves, the so-called Mori's breaking technique and results on uniform bundles) the following theorem. Let \(X\) be a Fano manifold of index \(r\) and dimension \(2r\) (characteristic 0). Assume \(r \geq 3\) and \(b_ 2(X) \geq 2\). Then \(X\) is either a projective bundle over a Fano manifold of dimension \(r + 1\) or it is a quadric bundle over a smooth variety or \(X\) has two elementary contractions such that either each contraction makes \(X\) a non- equidimensional scroll or one of the contractions makes \(X\) a non- equidimensional scroll and the other is birational and of divisorial type. In an added in proof the author describes a later result (a complete classification of Fano manifolds of index \(r\), dimension \(2r\) and \(b_ 2 \geq 2)\) contained in a joint paper of the reviewer and the author [``On Banica sheaves and Fano manifolds'', Compos. Math. (to appear)].
reviewed by: E. Ballico
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14E30
zbMATH DE Number: 567440
zbMATH Keywords: divisorial contraction
zbMATH Keywords: extremal ray contraction
zbMATH Keywords: Mori theory
zbMATH Keywords: deformations of rational curves
zbMATH Keywords: Fano manifold
zbMATH Keywords: quadric bundle
zbMATH Keywords: scroll
MaRDI profile type: Publication
cites work: Fibrés uniformes de rang élevé sur \(\mathbb{P}_ 2\)
cites work: Remarks on quasi-polarized varieties
cites work: Generalized adjunction and applications
cites work: Q3338372
cites work: Projective manifolds with ample tangent bundles
cites work: Biregular classification of Fano 3-folds and Fano manifolds of coindex 3
cites work: Vector bundles on complex projective spaces
cites work: Self maps of homogeneous spaces
cites work: Fano manifolds and vector bundles
cites work: On contractions of extremal rays of Fano manifolds.
cites work: On a conjecture of Mukai
cites work: On Fano manifolds of large index
cites work: Q4014762
cites work: On ample vector bundles whose adjunction bundles are not numerically effective
OpenAlex ID: W2105497016
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description: scientific article; zbMATH DE number 567440