Extending torsors over regular models of curves

Abstract

Let R be a discrete valuation ring with field of fractions K and residue field k of characteristic $p>0$. Given a commutative finite group scheme G over K and a smooth projective curve C over K with a rational point, we study the extension of pointed fppf G-torsors over C to pointed torsors over some R-regular model ${\mathcal {C}}$ of C. We first study this problem in the category of log schemes: given a finite flat R-group scheme ${\mathcal {G}}$, we prove that the data of a pointed ${\mathcal {G}}$-log torsor over ${\mathcal {C}}$ is equivalent to that of a morphism ${\mathcal {G}}^D \rightarrow {{\,\mathrm{Pic}\,}}^{log}_{{\mathcal {C}}/R}$, where ${\mathcal {G}}^D$ is the Cartier dual of ${\mathcal {G}}$ and ${{\,\mathrm{Pic}\,}}^{log}_{{\mathcal {C}}/R}$ the log Picard functor. After that, we give a sufficient condition for such a log extension to exist, and then we compute the obstruction for the existence of an extension in the category of usual schemes. In a second part, we generalize a result of Chiodo (Manuscr Math 129(3):337–368, 2009) which gives a criterion for the r-torsion subgroup of the Néron model of J to be a finite flat group scheme, and we combine it with the results of the first part. Finally, we give a detailed example of extension of torsors when C is a hyperelliptic curve defined over ${\mathbb {Q}}$, which illustrates our techniques.