Title: 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 $F$.