Mathieu Florence and Zinovy Reichstein: On the rationality problem for forms of moduli spaces of stable marked curves of positive genus ,

Submission: 2018, Jan 10

Let Mg, n (respectively, Mg, 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 Mg, n (or equivalently, Mg, 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 Mg, n is unirational for only finitely many pairs (g, n) with g ≥ 1. Finding the precise range of pairs (g, n), where Mg, 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 Mg, n defined over an arbitrary field F of characteristic different from 2. We show that all F-forms of Mg, 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

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