EXTENDING TORSORS UNDER QUASI-FINITE FLAT GROUP SCHEMES

Abstract Let R be a discrete valuation ring of field of fractions K and of residue field k of characteristic $p> 0$ . In an earlier work, we studied the question of extending torsors over K -curves into torsors over R -regular models of the curves in the case when the structural K -group scheme of the torsor admits a finite flat model over R . In this paper, we first give a simpler description of the problem in the case where the curve is semistable using recent work in Holmes, Molcho, Orecchia, and Poiret (2023, Journal für die Reine und Angewandte Mathematik [Crelle’s Journal] 230, 115–159) and Molcho and Wise (2022, Compositio Mathematica 158, 1477–1562). Second, if R is assumed to be Henselian and Japanese, we solve the problem of extending torsors by combining our previous work together with results in Antei and Emsalem (2018, Nagoya Mathematical Journal 230, 18–34) and Phung and Dos Santos (2023, Algebraic Geometry 230, 1–40), including the case where the structural group does not admit a finite flat R -model.