Appears in collection : Cohomological Methods in the Theory of Algebraic Groups

Let $\overline{M_{g,n}}$ be the moduli space of stable curves of genus $g$ with $n$ marked points. It is a classical problem in algebraic geometry to determine which of these spaces are rational over $\mathbb{C}$. In this talk, based on joint work with Mathieu Florence, I will address the rationality problem for twisted forms of $\overline{M_{g,n}}$ . Twisted forms of $\overline{M_{g,n}}$ are of interest because they shed light on the arithmetic geometry of $\overline{M_{g,n}}$, and because they are coarse moduli spaces for natural moduli problems in their own right. A classical result of Yu. I. Manin and P. Swinnerton-Dyer asserts that every form of $\overline{M_{0,5}}$ is rational. (Recall that the $F$-forms $\overline{M_{0,5}}$ are precisely the del Pezzo surfaces of degree 5 defined over $F$.) Mathieu Florence and I have proved the following generalization of this result. Let $ n\geq 5$ is an integer, and $F$ is an infinite field of characteristic $\neq$ 2. (a) If $ n$ is odd, then every twisted $F$-form of $\overline{M_{0,n}}$ is rational over $F$. (b) If $n$ is even, there exists a field extension $F/k$ and a twisted $F$-form of $\overline{M_{0,n}}$ which is unirational but not retract rational over $F$. We also have similar results for forms of $\overline{M_{g,n}}$ , where $g \leq 5$ (for small $n$ ). In the talk, I will survey the geometric results we need about $\overline{M_{g,n}}$ , explain how our problem reduces to the Noether problem for certain twisted goups, and how this Noether problem can (sometimes) be solved.

Keywords: rationality - moduli spaces of marked curves - Galois cohomology - Noether's problem