On a rationality problem for fields of cross-ratios II

Nghiem, Tran-Trung ; Reichstein, Zinovy

Canadian mathematical bulletin, Tome 64 (2021) no. 3, pp. 667-677

Résumé

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 $n \leqslant 4$.

Détail

DOI : 10.4153/S0008439520000739

Mots-clés : Rationality problem, field extension, cross ratio, conic curve