THE RATIONALITY PROBLEM FOR FORMS OF M0,n

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 F -forms of the moduli space M0,5 of stable curves of genus 0 with 5 marked points. Suppose n > 5 is an integer, and F is an infinite field of characteristic 6= 2. It is easy to see that every twisted F -form of M0,n is unirational over F . We show that (a) If n is odd, then every twisted F -form of M0,n is rational over F . (b) If n is even, there exists a field extension F/k and a twisted F -form X of M0,n such that X is not retract rational over F .