2017/09/17 by Mathieu Florence, Florence, Mathieu, Norbert Hoffmann +3
Computer Science · Mathematics · #14E08 #14G27 #14H10 #14H45 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1709.05696
openalex publication_date 2017/09/17 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
Let Mg, n (respectively, \Mg, n) be the moduli space of\nsmooth (respectively stable) curves of genus g with n marked points. Over\nthe field of complex numbers, it is a classical problem in algebraic geometry\nto determine whether or not Mg, n (or equivalently, \Mg, n)\nis a rational variety. Theorems of J. Harris, D. Mumford, D. Eisenbud and G.\nFarkas assert that Mg, n is not unirational for any n geqslant 0 if g\n geqslant 22. Moreover, P. Belorousski and A. Logan showed that Mg, n is\nunirational for only finitely many pairs (g, n) with g geqslant 1. Finding\nthe precise range of pairs (g, n), where Mg, n is rational, stably\nrational or unirational, is a problem of ongoing interest. In this paper we\naddress the rationality problem for twisted forms of \Mg, n\ndefined over an arbitrary field F of characteristic \≠ 2. We show that\nall F-forms of \Mg, n are stably rational for g = 1 and 3\n leqslant n leqslant 4, g = 2 and 2 leqslant n leqslant 3, g = 3 and\n1 leqslant n leqslant 14, g = 4 and 1 leqslant n leqslant 9, g = 5\nand 1 leqslant n leqslant 12.\n