Three theorems on existence of good quotients
type: Item
instance of: scholarly article
title: Three theorems on existence of good quotients
zbMATH Open document ID: 0870.14034
published in: Mathematische Annalen
publication date: 1997-09-17T00:00:00Z
review text: Let a reductive group \(G\) act on an algebraic variety \(X\). A \(G\)-morphism \(\pi:X\to Y\), where \(Y\) is an algebraic \(G\)-space with the trivial action of \(G\), is said to be a good quotient of \(G\), if \(\pi\) is affine and \(\pi_*({\mathcal O}_X)^G={\mathcal O}_Y\). The first of the three theorems mentioned in the title, says that if, for every curve \(C\) embedded in \(X\), there exists a good quotient of \(\overline{G}\overline{C}\) by \(G\) then there exists a good quotient of \(X\) by \(G\). -- The second theorem is, that if \(X\) is normal and there exists an affine \(G\)-morphism of \(X\) onto an algebraic space \(Y\) with the trivial action of \(G\), then there exists a good quotient of \(X\) by \(G\). -- The third theorem says that, if for any pair of points \((x,y)\) in \(X\), there exists their common \(G\)-invariant neighborhood \(U_{x,y}\) with a good quotient by \(G\), then there exists a good quotient of \(X\) by \(G\). Though the theorems have different character their proofs are related to each other.
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 1031524
zbMATH Keywords: action of reductive groups of an algebraic variety
zbMATH Keywords: good quotient
author: Andrzej Białynicki-Birula
author: Joanna Świȩcicka
MaRDI profile type: Publication
full work available at URL: s002080050027
OpenAlex ID: W1994159795
DOI: 10.1007/S002080050027
description: scientific article; zbMATH DE number 1031524