Rationality of some projective linear actions
type: Item
instance of: scholarly article
title: Rationality of some projective linear actions
zbMATH Open document ID: 0993.12003
author: Hamza Ahmad
author: Mowaffaq Hajja
author: Ming-Chang Kang
published in: Journal of Algebra
publication date: 2001-11-18T00:00:00Z
review text: Let \(K\) be a field, \(V\) a finite-dimensional vector space over \(K\), \(K[V]\) a coordinate ring of \(V\) over \(K\), and \(K(V)\) the function field of \(V\) over \(K\). Specifically, if \(x_1,x_2,\dots,x_n\) is a basis of \(V^*\), the dual space of \(V\), then \(K[V]= K[x_1,\dots,x_n]\), the polynomial ring of \(n\) variables over \(K\), and \(K(V)= K(x_1,\dots, x_n)\), the quotient field of \(K[x_1,\dots,x_n]\). Let \(P(V)\) denote the projective space associated with \(V\). Then define \(K(P(V))\) to be the subfield \(K(x_1/x_n,\dots,x_{n-1}/x_n)\) of \(K(V)\). Let \(G\) be a subgroup of \(GL(V)\). In this paper, the authors study under what conditions the fixed fields \(K(V)^G\) and \(K(P(V))^G\) are rational over \(K\). The authors do not restrict themselves to the assumptions that \(G\) is finite or that \(K\) is algebraically closed. The authors show the following result. Suppose that \(G=\langle\sigma\rangle\) and \(\dim_K V\leq 3\). Then both \(K(V)^G\) and \(K(P(V))^G\) are rational over \(K\). In addition, if \(G\) acts diagonally on \(V\oplus\cdots\oplus V\), then both \(K(V\oplus\cdots\oplus V)^G\) and \(K(P(V)\times\cdots\times P(V))^G\) are rational over \(K\). The authors also show the following result. If \(G=\langle \sigma\rangle\) acts on \(K(x_1,\dots,x_n,y_1,\dots,y_m)\) by \(\sigma(x_j)=(a_1x_j+b_1)/(c_1x_j+d_1)\) and \(\sigma(y_k)=(a_2y_k+b_2)/(c_2y_k+d_2)\), \(j=1,\dots,n\), \(k=1,\dots,m\), \(a_id_i-b_ic_i\neq 0\), \(a_i,b_i,c_i,d_i\in K\), \(i=1,2\), then \(K(x_1,\dots,x_n,y_1,\dots,y_m)^G\) is rational over \(K\).
reviewed by: John N. Mordeson
Mathematics Subject Classification ID: 12F20
Mathematics Subject Classification ID: 12F99
zbMATH DE Number: 1471834
zbMATH Keywords: rationality
zbMATH Keywords: group actions
zbMATH Keywords: fixed subfields
MaRDI profile type: Publication
full work available at URL: jabr.2000.8292
OpenAlex ID: W2047795913
cites work: Negligibility of projective linear automorphisms
cites work: Invariants of finite Abelian groups
cites work: Rationality of finite groups of monomial automorphisms of k(x,y)
cites work: A note on affine automorphisms
cites work: Finite group actions on rational function fields
cites work: Some actions of symmetric groups
cites work: RATIONALITY OF FIELDS OF INVARIANTS OF SOME FOUR-DIMENSIONAL LINEAR GROUPS, AND AN EQUIVARIANT CONSTRUCTION RELATED TO THE SEGRE CUBIC
cites work: Rational functions invariant under a finite Abelian group
cites work: Invariants of Certain Groups I
cites work: Q3140228
cites work: On subfields of rational function fields
cites work: Q4026923
cites work: Noether's problem over an algebraically closed field
cites work: Q3947744
cites work: Invariant rational functions and a problem of Steenrod
cites work: Q3324057
cites work: Rationality of the field of invariants of a triangular group
cites work: ON TWO-DIMENSIONAL ALGEBRAIC TORI. II
cites work: ON THE QUESTION OF THE STRUCTURE OF THE SUBFIELD OF INVARIANTS OF A CYCLIC GROUP OF AUTOMORPHISMS OF THE FIELDQ(x1,...,xn)
cites work: On Castelnuovo's criterion of rationality P\(_a\)=P\(_2\)=0 of an algebraic surface
DOI: 10.1006/JABR.2000.8292
description: scientific article; zbMATH DE number 1471834