Translator Disclaimer
June 2020 On a Rationality Problem for Fields of Cross-ratios
Zinovy REICHSTEIN
Tokyo J. Math. 43(1): 205-213 (June 2020). DOI: 10.3836/tjm/1502179305

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 $\{ 1, \dots, n \}$.

Citation

Download Citation

Zinovy REICHSTEIN. "On a Rationality Problem for Fields of Cross-ratios." Tokyo J. Math. 43 (1) 205 - 213, June 2020. https://doi.org/10.3836/tjm/1502179305

Information

Published: June 2020
First available in Project Euclid: 24 August 2019

zbMATH: 07227186
MathSciNet: MR4121794
Digital Object Identifier: 10.3836/tjm/1502179305

Subjects:
Primary: 14E08
Secondary: 12G05, 16H05, 16K50

Rights: Copyright © 2020 Publication Committee for the Tokyo Journal of Mathematics

JOURNAL ARTICLE
9 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.43 • No. 1 • June 2020
Back to Top