Noether's problem for transitive permutation groups of degree 6

Abstract

Suppose that a finite group $G$ is realized in the Cremona group $\mathrm{Cr}_m (k)$, the group of $k$-automorphisms of the rational function field $K$ of $m$ variables over a constant field $k$. The most general version of Noether's problem is then to ask, whether the subfield $K^G$ consisting of $G$-invariant elements is again rational or not. This paper treats Noether's problem for various subgroups $G$ of $\mathfrak{S}_6$, the symmetric group of degree 6, acting on the function field $\boldsymbol{Q}(s, t, z)$ over $k = \boldsymbol{Q}$ of the moduli space $\mathcal{M}_{0,6}$ of $\mathbb{P}^1$ with ordered six marked points. We shall show that this version of Noether's problem has an affirmative answer for all but two conjugacy classes of transitive subgroups $G$ of $\mathfrak{S}_6$, by exhibiting explicitly a system of generators of the fixed field $\boldsymbol{Q}(s,t,z)^G$. In the exceptional cases $G \cong \mathfrak{A}_6, \mathfrak{A}_5$, the problem remains open.