Extension of finite solvable torsors over a curve

Abstract

Let R be a 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 with geometrically connected fibers endowed with a section x ∈ X(R). Let G be a finite solvable K-group scheme and assume that either |G| = p n or G has a normal series of length 2. We prove that every quotient pointed G-torsor over the generic fiber X η of X can be extended to a torsor over X after eventually extending scalars and after eventually blowing up X at a closed subscheme of its special fiber X s .Mathematics Subject Classification: 14H30, 14L15.

Models of Torsors Over Affine Spaces

“…So far the problem of extending the G-torsor Y → X η has consisted in finding a finite and flat S-group scheme G ′ whose generic fibre is isomorphic to G and a G ′ -torsor T → X whose generic fibre is isomorphic to Y → X η as a G-torsor. Some solutions, from Grothendieck's first ideas until nowadays, are known in some particular relevant cases and are the object of many classical and well known results and more recent papers, see for instance [11,Exposé X], [17, §3], [19, §2.4], [20,Corollary 4.2.8], [3] and [2]. However a general solution does not exist.…”

Models of Torsors and the Fundamental Group Scheme

“…Some solutions to this problem, from Grothendieck's first ideas until nowadays, are known in some particular relevant cases that we briefly recall: Grothendieck proved that, possibly after extending scalars, the problem has a solution when G is a constant finite group, S is the spectrum of a complete discrete valuation ring with algebraically closed residue field of positive characteristic p, with X proper and smooth over S with geometrically connected fibers and p ∤ |G| ( [10], Exposé X); when S is the spectrum of a discrete valuation ring of residue characteristic p, X is a proper and smooth curve over S then Raynaud suggested a solution, possibly after extending scalars, for |G| = p ( [19], §3); when S is the spectrum of a discrete valuation ring R of mixed characteristic (0, p) Tossici provided a solution, possibly after extending scalars, for G commutative when X is a regular scheme, faithfully flat over S, with some extra assumptions on X and Y ( [21], Corollary 4.2.8). Finally in [4], §3.2 and §3.3 the first author provided a solution for G commutative, when S is a connected Dedekind scheme and f : X → S is a smooth morphism satisfying additional assumptions (in this last case it is not needed to extend scalars) and in [3] the case where G is solvable is treated. However a general solution does not exist.…”

“…Finally, in [2, Sections 3.2 and 3.3], the first author provided a solution for commutative, when is a connected Dedekind scheme and is a smooth morphism satisfying additional assumptions (in this last case it is not needed to extend scalars). In [3], the case where is solvable is treated. However, a general solution does not exist.…”