On a rationality problem for fields of cross-ratios

TL;DR

This paper investigates the rationality of fixed fields under group actions related to cross-ratios, establishing equivalences between rationality, unirationality, and orbit properties of subgroups within the symmetric group.

Contribution

It extends Tsunogai's results by framing the problem in Galois cohomology and proves the equivalence of rationality, unirationality, and orbit conditions for subgroups.

Findings

Rationality over $K_n^S$ is equivalent to having an odd order orbit.

The problem is recast in terms of Galois cohomology.

Main theorem links subgroup orbit properties to field rationality.

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 $S_n$ 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 $L_n^{{\rm PGL}_2}$ is called "the field of cross-ratios". Let $S \subset S_n$ be a subgroup. The Noether Problem asks whether the field extension $L_n^S/k$ 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 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 $K_n^S$, (c) $S$ has an orbit of odd order in $\{1, \dots, n \}$.