On a rationality problem for fields of cross-ratios II

Abstract

Let $k$ be a field, $x_1,\dots ,x_n$ be independent variables and $L_n=k(x_1,\dots ,x_n)$. The symmetric group $\hat{I}{\pounds }_n$ acts on $L_n$ by permuting the variables, and the projective linear group ${\text{PGL}}_2$ acts by \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \colon x_i \mapsto \frac{a x_i + b}{c x_i + d} \] for each $i=1,\dots ,n$. The fixed field $L_n^{{\text{PGL}}_2}$ is called "the field of cross-ratios". Given a subgroup $S\subset \hat{I}{\pounds }_n$, H. Tsunogai asked whether $L_n^S$ rational over $K_n^S$. When $n⩾5$ the second author has shown that $L_n^S$ is rational over $K_n^S$ if and only if $S$ has an orbit of odd order in $\{1,\dots ,n\}$. In this paper we answer Tsunogai's question for $n⩽4$.

9 pages; to be appeared on the Canadian Mathematical Bulletin