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

Submission: 2017, Jun 17, revised: 2018, Jan 14

A theorem of Yu. I. Manin and P. Swinnerton-Dyer asserts that every del
Pezzo surface of degree 5 over an arbitrary field F is rational. In this
paper we generalize this result as follows. Recall that del Pezzo
surfaces of degree 5 over F are precisely the F-forms of the moduli space
** M** _{0, n} 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, 14H10, 20G15, 16K50 16K50

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

Full text: dvi.gz 31 k, dvi 65 k, ps.gz 969 k, pdf.gz 177 k, pdf 201 k.

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