Mathieu Florence and Zinovy Reichstein: The rationality problem for forms of *M*_{o, n}

mathieu.florence@gmail.com, reichst@math.ubc.ca

Submission: 2017, Jun 17

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 *M*_{0,5} of stable curves of genus 0 with 5 marked points. Suppose n >= 5 is an integer, and F is an infinite field of characteristic different from 2. It is easy to see that every twisted F-form of *M*_{0, n} is unirational over F. We show that (a) If n is odd, then every twisted F-form of*M*_{0, n} is rational over F. (b) If n is even, there exists a field extension E/F and twisted E-form X of *M*_{0, n} such that X is not retract rational over E.

2010 Mathematics Subject Classification: 14E08, 14M20, 14H10, 20G15, 16K50

Keywords and Phrases: Rationality, moduli spaces of marked curves, Galois cohomology, Brauer group

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