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Extending torsors under quasi-finite flat group schemes

Sara Mehidi

TL;DR

The paper addresses extending torsors on $K$-curves to $R$-models, first providing a streamlined treatment for semistable curves via the log Picard framework, then tackling the case where the torsor’s structural group lacks a finite flat $R$-model by leveraging quasi-finite flat models. The main method blends log-geometry (Kummer log flat torsors, divisorial log structures) with classical objects like the log Picard functor and the Néron model, yielding concrete extension criteria: a torsor extends to a log torsor precisely when the associated dual-group morphism extends to the Néron model, and, for finite subgroups of Jacobians, a finite-flat closure condition governs extendability. In Part II, the work integrates Antei’s model-extension results with Pedro’s descent to finite-flat cases, establishing existence results for extensions under quasi-finite flat group schemes and showing how to reduce to finite-flat substructures. Overall, the results deepen the understanding of how torsors over $K$-curves extend over $R$-models in both semistable and generalized quasi-finite settings, with implications for ramification-aware extensions and the construction of compatible log-structures.

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 on $K$-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. Secondly, if $R$ is assumed to be Henselian and Japanese, we solve the problem of extending torsors even if the structural group does not admit a finite flat $R$-model.

Extending torsors under quasi-finite flat group schemes

TL;DR

The paper addresses extending torsors on -curves to -models, first providing a streamlined treatment for semistable curves via the log Picard framework, then tackling the case where the torsor’s structural group lacks a finite flat -model by leveraging quasi-finite flat models. The main method blends log-geometry (Kummer log flat torsors, divisorial log structures) with classical objects like the log Picard functor and the Néron model, yielding concrete extension criteria: a torsor extends to a log torsor precisely when the associated dual-group morphism extends to the Néron model, and, for finite subgroups of Jacobians, a finite-flat closure condition governs extendability. In Part II, the work integrates Antei’s model-extension results with Pedro’s descent to finite-flat cases, establishing existence results for extensions under quasi-finite flat group schemes and showing how to reduce to finite-flat substructures. Overall, the results deepen the understanding of how torsors over -curves extend over -models in both semistable and generalized quasi-finite settings, with implications for ramification-aware extensions and the construction of compatible log-structures.

Abstract

Let be a discrete valuation ring of field of fractions and of residue field of characteristic . In an earlier work, we studied the question of extending torsors on -curves into torsors over -regular models of the curves in the case when the structural -group scheme of the torsor admits a finite flat model over . In this paper, we first give a simpler description of the problem in the case where the curve is semistable. Secondly, if is assumed to be Henselian and Japanese, we solve the problem of extending torsors even if the structural group does not admit a finite flat -model.
Paper Structure (13 sections, 11 theorems, 8 equations)

This paper contains 13 sections, 11 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kummer log flat torsors
  4. Log schemes.
  5. Charts.
  6. Inverse image log structure and strict morphisms.
  7. Direct image log structure.
  8. Extension of torsors under a finite flat group scheme
  9. Part I: Case of semistable curves
  10. The Log Picard functor
  11. On the existence of a finite flat model of the group scheme
  12. Part II: Extension of torsors under a quasi-finite flat group scheme
  13. Acknowledgment

Key Result

Theorem 2.4

Sara Let $C$ be a smooth projective and geometrically connected curve over $K$, endowed with a $K$-point $Q$, and let $J$ denote its Jacobian variety. Let $\mathcal{C}$ be an $R$-regular model of $C$ such that its special fiber is a normal crossing divisor. Endow $\mathcal{C}$ with the divisorial lo where $H^1_{klf}(\mathcal{C},\mathcal{Q},\mathcal{G})$ denotes the cohomology group that classifies

Theorems & Definitions (21)

  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Theorem 3.1
  • Remark 3.2
  • Proposition 3.3
  • Proof
  • Corollary 3.4
  • ...and 11 more