G-torsors and universal torsors over nonsplit del Pezzo surfaces

2021/09/16 by Derenthal, Ulrich, Hoffmann, Norbert
#11E57 #14G27 (Secondary) #14J26 (Primary) #14L30 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2109.08137

Abstract

Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of S_L in Pic S_L, or a form of it containing the Néron-Severi torus. Let G be the G-torsor over S_L obtained by extension of structure group from a universal torsor T over S_L. We prove that G does not descend to S unless T does. This is in contrast to a result of Friedman and Morgan that such G always descend to singular del Pezzo surfaces over ℂ from their desingularizations.

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