Rationality questions for forms of moduli spaces, I

This talk is about joint work with Zinovy Reichstein. Let $F$ be a field. Denote by $\mathcal M_{0,n}$ the moduli space of (say, smooth) curves of genus zero, equipped with $n > 4$ distinct marked points. As an $F$-variety, it is $F$-rational, of dimension $n-3$. One can consider its twisted forms, that is to say, the $F$-varieties $X$, becoming isomorphic to $\mathcal M_{0,n}$ over an algebraic closure of $F$. Our main theorem then goes a follows. If $n > 5$ is odd, then all $F$-forms of $\mathcal M_{0,n}$ are $F$-rational. If $n > 6$ is even, then there exists (over some $F$) non $F$-retract rational such $F$-forms. Note that $F$-forms of $\mathcal M_{0,5}$ are exactly del Pezzo surfaces of degree $5$. Hence, the positive part of our theorem generalizes a well-known result of Manin and Swinnerton-Dyer, asserting that Del Pezzo surfaces of degree $5$ are rational. I shall discuss the various techniques involved in the proof, strongly related to Noether’s problem.