mathieu.florence@imj-prg.fr , reichst@math.ubc.ca

Submission: 2018, Jan 10

Let M_{g, n} (respectively, M_{g, n} ) be the moduli space of smooth (respectively stable) curves of genus g with n marked points. Over the field of complex numbers, it is a classical problem in algebraic geometry to determine whether or not M_{g, n} (or equivalently, M_{g, n} ) is a rational variety. Theorems of J. Harris, D. Mumford, D. Eisenbud and G. Farkas assert that M _{g, n} is not unirational for any positive integer n, as long as g ≥ 22. Moreover, P. Belorousski and A. Logan showed that M_{g, n} is unirational for only finitely many pairs (g, n) with g ≥ 1. Finding the precise range of pairs (g, n), where M_{g, n} is rational, stably rational or unirational, is a problem of ongoing interest. In this paper we address the rationality problem for twisted forms of M_{g, n} defined over an arbitrary field F of characteristic different from 2. We show that all F-forms of M_{g, n} are stably rational for g = 1 and 3 ≤ n ≤ 4, g = 2 and 2 ≤ n ≤ 3, g = 3 and 1 ≤ n ≤ 14, g = 4 and 1 ≤ n ≤ 9, g = 5 and 1 ≤ n ≤ 12.

2010 Mathematics Subject Classification: 14E08, 14H10, 14G27, 14H45

Keywords and Phrases: Stable curves, marked curves, moduli spaces, rationality problem, group actions, twisted varieties, Weil restriction

Full text: dvi.gz 66 k, dvi 165 k, ps.gz 983 k, pdf.gz 293 k, pdf 326 k.