Spherical varieties and Mori theory
type: Item
instance of: scholarly article
title: Spherical varieties and Mori theory
zbMATH Open document ID: 0821.14029
DOI: 10.1215/S0012-7094-93-07213-4
author: Michel Brion
published in: Duke Mathematical Journal
publication date: 1995-10-03T00:00:00Z
review text: This article contains many interesting results on \(B\)-stable curves on normal \(G\)-varieties. For example: In a projective spherical variety each irreducible \(B\)-stable curve is isomorphic to the projective line. Here \(G\) is a connected reductive algebraic group (over an algebraically closed field of characteristic 0) and \(B\) is a Borel subgroup of \(G\). A normal algebraic variety with \(G\)-action is spherical if \(B\) contains a dense orbit. A spherical variety is uniquely determined by its ``coloured fan'' (certain combinatorial data). The main objective of this paper is to study spherical varieties from the viewpoint of Mori theory. [Thus results of \textit{M. Reid} in: Arithmetic and geometry, Vol. II, Prog. Math. 36, 395-418 (1983; Zbl 0571.14020) for toric varieties are generalized.] For a complete spherical variety \(X\) the author proves the following: \(\text{NE}(X)\), the cone of effective 1-cycles, is polyhedral and generated by the classes of \(B\)-stable rational curves. A system of generators is described in terms of the coloured fan of \(X\). Moreover, flips in spherical \(\mathbb{Q}\)-factorial varieties are studied.
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14M17
zbMATH DE Number: 558967
zbMATH Keywords: Néron model
zbMATH Keywords: stable curve
zbMATH Keywords: coloured fan
zbMATH Keywords: spherical variety
zbMATH Keywords: Mori theory
zbMATH Keywords: effective 1-cycles
zbMATH Keywords: flips
reviewed by: Vyacheslav V. Shokurov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 558967