Rationality of three-dimensional quotients by monomial actions (Q618242)

Let \(K\) be a field and let \(K(x_1,\ldots ,x_n)\) denote the rational function field over \(K\) in \(n\) indeterminates. A \(K\)-automorphism \(\sigma\) of \(K(x_1,\ldots ,x_n)\) is called \textit{monomial} if, for each \(1\leq j \leq n\), \[ \sigma(x_j)=a_j(\sigma)\prod_{1\leq i\leq n}x_i^{m_{i,j}} , \] for some \(a_j(\sigma)\in K\backslash \{0\}\) and \((m_{i,j})_{1\leq i,j \leq n} \in \text{GL}_n(\mathbb{Z})\). The paper under review deals with the following type of rationality problem: given a finite group \(G\) acting on \(K(x_1,\ldots ,x_n)\) by monomial \(K\)-automorphisms, is the fixed field \(K(x_1,\ldots ,x_n)^G\) purely transcendental over \(K\)? In the case \(n=2\), this question was answered in the affirmative, for every finite group \(G\) and every field \(K\), by \textit{M. Hajja} in [J. Algebra 85, No. 1, 243--250 (1983; Zbl 0519.12016)] and [J. Algebra 109, No. 1, 46--51 (1987; Zbl 0624.12014)]. In the case \(n=3\), the same conclusion holds for \textit{purely monomial} actions (meaning that \(a_j(\sigma)=1\) for every \(\sigma \in G\) and \(1\leq j \leq n\)), from work by \textit{M. Hajja} and \textit{M. Kang} [J. Algebra 149, No. 1, 139--154 (1992; Zbl 0760.12004)] and [J. Algebra 170, No. 3, 805--860 (1994; Zbl 0831.14003)], and by \textit{A. Hoshi} and \textit{Y. Rikuna} [Math. Comput. 77, No. 263, 1823--1829 (2008; Zbl 1196.14018)]. On the other hand, there exist examples of non-pure monomial actions such that \(K(x_1,x_2,x_3)^G\) is not \(K\)-rational, nor even retract-rational. The main result of this paper reads as follows: Let \(K\) be a quadratically closed field of characteristic \(\text{char}(K)\neq 2\) and let \(G\) be a finite \(2\)-group. If \(G\) acts on \(K(x_1,x_2,x_3)\) by monomial \(K\)-automorphisms, then \(K(x_1,x_2,x_3)^G\) is purely transcendental over \(K\). In order to prove this, the authors show first that it suffices to consider the case in which the homomorphism \(G\rightarrow \text{GL}_3(\mathbb{Z})\), \(\sigma \mapsto (m_{i,j})_{1\leq i,j \leq n}\), is injective. Then, they examine the \(36\) different cases corresponding to the conjugacy classes of finite subgroups of \(\text{GL}_3(\mathbb{Z})\) which are \(2\)-groups. In each case, suitable computations allow the authors to reduce the proof to some known rationality results obtained by several authors, including A. Ahmad, H. Chu, M. Hajja, S.-J. Hu, M. Kang and J. Ohm. An alternative geometric proof is also given, under the extra assumption that \(K\) is an algebraically closed field of characteristic \(\text{char}(K)=0\). In addition, the authors obtain some results concerning the Linear Noether's Problem, as an application of their main result.