Special groups, versality and the Grothendieck-Serre conjecture

2019/12/17 by Reichstein, Zinovy, Tossici, Dajano · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1912.08109

Abstract

Let k be a base field and G be an algebraic group over k. J.-P. Serre defined G to be special if every G-torsor T → X is locally trivial in the Zariski topology for every reduced algebraic variety X defined over k. In recent papers an a priori weaker condition is used: G is called special if every G-torsor T → Spec(K) is split for every field K containing k. We show that these two definitions are equivalent. We also generalize this fact and propose a strengthened version of the Grothendieck-Serre conjecture based on the notion of essential dimension.

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