On a Rationality Problem for Fields of Cross-ratios

Abstract

Let $k$ be a field, $n \geqslant 5$ be an integer, $x_1, \dots, x_n$ be independent variables and $L_n = k(x_1, \dots, x_n)$. The symmetric group $\Sigma_n$ acts on $L_n$ by permuting the variables, and the projective linear group $\text{PGL}_2$ acts by applying (the same) fractional linear transformation to each variable. The fixed field $K_n = L_n^{\text{PGL}_2}$ is called ``the field of cross-ratios''. Let $S \subset \Sigma_n$ be a subgroup. The Noether Problem asks whether the field extension $L_n^S/k$ is rational, and the Noether Problem for cross-ratios asks whether $K_n^S/k$ 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 interesting situations, in particular in the case where $S = \Sigma_n$. 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 $S \subset \Sigma_n$ are equivalent: (a) $L_n^S$ is rational over $K_n^S$, (b) $L_n^S$ is unirational over $K_n^S$, (c) $S$ has an orbit of odd order in $\{ 1, \dots, n \}$.