Extremal rays on higher dimensional projective varieties
type: Item
instance of: scholarly article
title: Extremal rays on higher dimensional projective varieties
zbMATH Open document ID: 0762.14009
DOI: 10.1007/BF01445224
author: Qi Zhang
published in: Mathematische Annalen
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/164883
review text: The paper under review investigates small extremal contractions \(f\) of a smooth projective variety \(X\) of dimension \(n=2k\) under the following assumption. Suppose that each irreducible component \(E_ i\) of the exceptional locus \(E\) of \(f\) is a smooth subvariety of dimension \(k\). Theorem 1 states that then \(E_ i\cong\mathbb{P}^ k\). Note that always, by \textit{J. A. Wiśniewski} [J. Reine Angew. Math. 417, 141-157 (1991; Zbl 0721.14023)], \(\dim E\geq k\). So, the considered case is extreme. But the assumption that \(E_ i\)'s are smooth is very strong. Under the last assumption and for \(X\) of dimension 5, the author proves that \(E_ i\cong\mathbb{P}^ 3\), or \(\mathbb{Q}^ 3\) (a 3-dimensional hyperquadric), or a linear \(\mathbb{P}^ 2\)-bundle over a smooth curve.
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14J40
zbMATH DE Number: 59967
zbMATH Keywords: extremal ray
zbMATH Keywords: Mori theory
zbMATH Keywords: minimal model program
zbMATH Keywords: small extremal contractions
reviewed by: Vyacheslav V. Shokurov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 59967