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

Abstract: 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 $n \geqslant 0$ if $g \geqslant 22$. Moreover, P. Belorousski and A. Logan showed that $M_{g, n}$ is unirational for only finitely many pairs $(g, n)$ with $g \geqslant 1$. 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 $\neq 2$. 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 $1 \leqslant n \leqslant 12$.