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On a rationality problem for fields of cross-ratios II

Part of: Special varieties Birational geometry

Published online by Cambridge University Press:  04 September 2020

Tran-Trung Nghiem  and
Zinovy Reichstein
Tran-Trung Nghiem
Affiliation:
Département de Mathématiques et Applications, École Normale Supérieure Paris, Paris, Francee-mail:tran-trung.nghiem@ens.fr
Zinovy Reichstein*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BCV6T 1Z2, Canada
*
e-mail: reichst@math.ubc.ca
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Abstract

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Let k be a field, $x_1, \dots , x_n$ be independent variables and let $L_n = k(x_1, \dots , x_n)$ . The symmetric group $\operatorname {\Sigma }_n$ acts on $L_n$ by permuting the variables, and the projective linear group $\operatorname {PGL}_2$ acts by

$$ \begin{align*} \begin{pmatrix} a & b \\ c & d \end{pmatrix}\, \colon x_i \longmapsto \frac{a x_i + b}{c x_i + d} \end{align*} $$
for each $i = 1, \dots , n$ . The fixed field $L_n^{\operatorname {PGL}_2}$ is called “the field of cross-ratios”. Given a subgroup $S \subset \operatorname {\Sigma }_n$ , 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 $n \leqslant 4$ .

Keywords

Rationality problemfield extensioncross ratioconic curve

MSC classification

Primary: 14E08: Rationality questions
Secondary: 14M17: Homogeneous spaces and generalizations
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 3 , September 2021 , pp. 667 - 677
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Copyright
© Canadian Mathematical Society 2020

Footnotes

Tran-Trung Nghiem was supported by a Fondation Hadamard Scholarship. Zinovy Reichstein was partially supported by National Sciences and Engineering Research Council of Canada Discovery grant 253424-2017.

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