Rationality of Some Projective Linear Actions

Abstract

Let K be any field which may not be algebraically closed, V a finite-dimensional Ž . vector space over K, g GL V where the order of can be finite or infinite.Ž . ² : Ž Ž .. ² : THEOREM. If dim V F 3, then both K V and K ‫ސ‬ V are rational K Ž . s purely transcendental over K. Similar results hold for a cyclic affine action.

Rationality of the quotient of ℙ2 by finite group of automorphisms over arbitrary field of characteristic zero

“…It corresponds to k-rationality of quotients of toric surfaces by groups having an invariant two-dimensional torus on such a surface. From results of the paper [1] it follows that a quotient P 2 k /G and a quotient P 1…”

Noether's problem for groups of order 32

“…. , z n ) so that σ (z i ) = z i for any σ ∈ G, any 1 i n. [AHK,Theorem 3.1].) Let L be any field, L(x) the rational function field of one variable over L, and G a finite group acting on L(x).…”

“…(See E. Noether, 1916 [AHK,Theorem 3.4].) Let K be any field, K (x, y) the rational function field of two variables over K , and G any group acting on K (x, y) by K -automorphisms.…”

Rationality problem of three-dimensional purely monomial group actions: the last case

“…Then we can check k(t 1 [7] and Masuda [11]. We omit displaying them because of their complicated expressions.…”