Models of torsors over curves

2012/10/04 by Marco Antei, Antei, Marco
Mathematics · #14L15 (Primary) 11G99 (Secondary) #14L30 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1210.1522

openalex publication_date 2012/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R be a complete discrete valuation ring with fraction field K and with algebraically closed residue field. Let X be a faithfully flat R-scheme of finite type of relative dimension 1 and G be any affine K-group scheme of finite type. We prove that every G-torsor Y over the generic fibre X_η of X can be extended to a torsor over X' under the action of an affine and flat K-group scheme of finite type G' where X' is obtained by X after a finite number of Néron blowing ups. Moreover if G is finite and étale (resp. admits a finite and flat model) we find X' such that G' is finite and étale (resp. finite and flat) after, if necessary, extending scalars. We provide examples explaining the new techniques.

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