Extension of finite solvable torsors over a curve

Abstract

Let R be a complete discrete valuation ring with fraction field K and with algebraically closed residue field of positive characteristic p. Let X be a smooth fibered surface over R. Let G be a finite, ,tale and solvable K-group scheme and assume that either |G| = p (n) or G has a normal series of length 2. We prove that for every connected and pointed G-torsor Y over the generic fibre of X there exist a regular fibered surface over R and a model map such that Y can be extended to a torsor over possibly after extending scalars.