Three-dimensional purely quasimonomial actions

Abstract

Let $G$ be a finite subgroup of ${Aut}_k\left(K\right(x_1,\dots ,x_n\left)\right)$, where $K/k$ is a finite field extension and $K(x_1,\dots ,x_n)$ is the rational function field with $n$ variables over $K$. The action of $G$ on $K(x_1,\dots ,x_n)$ is called quasimonomial if it satisfies the following three conditions: (i) $\sigma \left(K\right)\subset K$ for any $\sigma \in G$; (ii) $K^G=k$, where $K^G$ is the fixed field under the action of $G$; (iii) for any $\sigma \in G$ and $1\le j\le n$, $\sigma \left(x_j\right)=c_j\left(\sigma \right){\prod }_{i=1}^n{x}_i^{a_{ij}}$, where $c_j\left(\sigma \right)\in K^{\times }$ and $[a_{i,j}{]}_{1\le i,j\le n}\in {GL}_n(\mathbb{Z})$. A quasimonomial action is called purely quasimonomial if $c_j\left(\sigma \right)=1$ for any $\sigma \in G$ and any $1\le j\le n$. When $k=K$, a quasimonomial action is called monomial. The main question is: Under what situations is $K(x_1,\dots ,x_n{)}^G$ rational (i.e., $=$ purely transcendental) over $k$? For $n=1$, the rationality problem was solved by Hoshi, Kang, and Kitayama. For $n=2$, the problem was solved by Hajja when the action is monomial, by Voskresenskii when the action is faithful on $K$ and purely quasimonomial, which is equivalent to the rationality problem of $n$-dimensional algebraic $k$-tori which split over $K$, and by Hoshi, Kang, and Kitayama when the action is purely quasimonomial. For $n=3$, the problem was solved by Hajja, Kang, Hoshi, and Rikuna when the action is purely monomial, by Hoshi, Kitayama, and Yamasaki when the action is monomial except for one case, and by Kunyavskii when the action is faithful on $K$ and purely quasimonomial. In this paper, we determine the rationality when $n=3$ and the action is purely quasimonomial except for a few cases using a conjugacy classes move technique. As an application, we will show the rationality of some $5$-dimensional purely monomial actions which are decomposable.