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 $n⩾0$ if $g⩾22$. Moreover, P. Belorousski and A. Logan showed that $M_{g,n}$ 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 $\ne 2$. We show that all $F$-forms of $\overline{M_{g,n}}$ are stably rational for $g=1$ and $3⩽n⩽4$, $g=2$ and $2⩽n⩽3$, $g=3$ and $1⩽n⩽14$, $g=4$ and $1⩽n⩽9$, $g=5$ and $1⩽n⩽12$.

13 pages, proofs much shortened, new coauthor