Orbits and invariants of visible group actions
type: Item
instance of: scholarly article
title: Orbits and invariants of visible group actions
zbMATH Open document ID: 1284.14058
author: Lex E. Renner
published in: Transformation Groups
publication date: 2013-02-18T00:00:00Z
review text: Let \(X\) be an irreducible affine algebraic variety endowed with a regular action of an affine algebraic group \(G\). The author calls such an action visible if the field of \(G\)-invariant rational functions on \(X\) is the field of fractions of the algebra of \(G\)-invariant regular functions on \(X\). It is shown that if the action is not visible, then there is a visible action of \(G\) on a variety \(Z\) and a \(G\)-equivariant birational morphism \(Z\to X\). This is used for obtaining visible \(G\)-invariant open subsets of \(X\). Visibility is also explored in the presence of the following properties: (1) stability of the actions of \(G\) on \(X\), (2) the existence of a regular semi-invariant \(f\) such that the \(G\)-action on \(X_f\) is visible; (3) reductivity of \(G\).
Mathematics Subject Classification ID: 14L17
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 22F50
zbMATH DE Number: 6136957
zbMATH Keywords: affine algebraic group
zbMATH Keywords: action
zbMATH Keywords: semi-invariant
zbMATH Keywords: morphism
reviewed by: Vladimir L. Popov
MaRDI profile type: Publication
full work available at URL: s00031-012-9198-1
OpenAlex ID: W2004537896
cites work: Computing invariants of algebraic groups in arbitrary characteristic
cites work: Q5465577
cites work: Localization and invariant theory
cites work: Algebraic homogeneous spaces and invariant theory
cites work: Geometric Invariant Theory
cites work: Period, index and potential \(\text Ш\)
cites work: Q3506242
cites work: Observability in invariant theory
cites work: Observable actions of algebraic groups
cites work: Some Basic Theorems on Algebraic Groups
cites work: Q5735629
DOI: 10.1007/S00031-012-9198-1
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description: scientific article; zbMATH DE number 6136957