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

TL;DR

This paper investigates the rationality of twisted forms of moduli spaces of stable marked curves of positive genus over arbitrary fields, establishing stable rationality for specific genus and marked point combinations.

Contribution

It extends the understanding of rationality properties to all forms of moduli spaces over arbitrary fields, identifying stable rationality in several cases.

Findings

All forms of ar{M}_{g,n} are stably rational for specified (g,n) ranges.

Addresses rationality over arbitrary fields, not just complex numbers.

Provides new stable rationality results for moduli spaces 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 $1 \leqslant n \leqslant 12$.