On a rationality problem for fields of cross-ratios II

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 $Σ_n$ 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 $i = 1, \ldots, n$. The fixed field $L_n^{\text{PGL}_2}$ is called "the field of cross-ratios". Given a subgroup $S \subset Σ_n$, H. Tsunogai asked whether $L_n^S$ rational over $K_n^S$. When $n \geqslant 5$ 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 $n \leqslant 4$.

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