On a rationality problem for fields of cross-ratios

Let $k$ be a field, $n⩾5$ be an integer, $x_1,\dots ,x_n$ be independent variables and $L_n=k(x_1,\dots ,x_n)$. The symmetric group $S_n$ acts on $L_n$ by permuting the variables, and the projective linear group ${\rm PGL}_2$ acts by applying (the same) fractional linear transformation to each varaible. The fixed field $L_n^{{\mathrm{P}\mathrm{G}\mathrm{L}}_2}$ is called “the field of cross-ratios”. Let $S\subset S_n$ be a subgroup. The Noether Problem asks whether the field extension $L_n^S\mathrm{/}k$ is rational, and the Noether Problem for cross-ratios asks whether $K_n^S\mathrm{/}k$ is rational. In an effort to relate these two problems, H. Tsunogai posed the following question: Is $L_n^S$ rational over $K_n^S$? He answered this question in several situations, in particular, in the case where $S=S_n$. In this paper we extend his results by recasting the problem in terms of Galois cohomology. Our main theorem asserts that the following conditions on a subgroup $S\subset S_n$ are equivalent: (a) $L_n^S$ is rational over $K_n^S$, (b) $L_n^S$ is unirational over $K_n^S$, © $S$ has an orbit of odd order in $\{1,\dots ,n\}$.