On the rationality problem for forms of moduli spaces of stable marked curves of positive genus

Let $M_{g,n}$ (respectively, $\overline{M_{g,n}}$) be the moduli space of smooth (respectively stable) curves of genus $g$ with $n$ marked points. Over the field of complex numbers, it is a classical problem in algebraic geometry to determine whether or not $M_{g,n}$ (or equivalently, $\overline{M_{g,n}}$) is a rational variety. Theorems of J. Harris, D. Mumford, D. Eisenbud and G. Farkas assert that $M_{g,n}$ is not unirational for any $n⩾0$ if $g \geqslant 22$\mathrm{.}Moreover,P\mathrm{.}BelorousskiandA\mathrm{.}Loganshowedthat$M_{g, n}$ is unirational for only finitely many pairs $(g,n)$ with $g⩾1$. Finding the precise range of pairs $(g,n)$, where $M_{g,n}$ is rational, stably rational or unirational, is a problem of ongoing interest. In this paper we address the rationality problem for twisted forms of $\overline{M_{g,n}}$ defined over an arbitrary field $F$ of characteristic $\mathrm{\ne }2$. We show that all $F$-forms of $\overline{M_{g,n}}$ are stably rational for $g=1$ and $3 \leqslant n \leqslant 4$,$g = 2$and$2 \leqslant n \leqslant 3$,$g = 3$and$1 \leqslant n \leqslant 14$,$g = 4$and$1 \leqslant n \leqslant 9$,$g = 5$and$1 \leqslant n \leqslant 12$.