The rationality problem for forms of $\overline{M_{0, n}}$

Abstract: Let $X$ be a del Pezzo surface of degree $5$ defined over a field $F$. A theorem of Yu. I. Manin and P. Swinnerton-Dyer asserts that every Del Pezzo surface of degree $5$ is rational. In this paper we generalize this result as follows. Recall that del Pezzo surfaces of degree $5$ over a field $F$ are precisely the twisted $F$-forms of the moduli space $\overline{M_{0, 5}}$ of stable curves of genus $0$ with $5$ marked points. Suppose $n \geq 5$ is an integer, and $F$ is an infinite field of characteristic $\neq 2$. It is easy to see that every twisted $F$-form of $\overline{M_{0, n}}$ is unirational over $F$. We show that (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 $X$ of $\overline{M_{0, n}}$ such that $X$ is not retract rational over $F$.