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arxiv: 2607.02014 · v1 · pith:75ALVCEGnew · submitted 2026-07-02 · 🧮 math.AG

Comb smoothing and local triviality of homogeneous spaces over a relative curve

Margot Bruneaux , Federico Scavia This is my paper

Pith reviewed 2026-07-03 06:00 UTC · model grok-4.3

classification 🧮 math.AG
keywords reductive groupsG-torsorslocal trivialitycomb smoothingHenselian ringshomogeneous spacesBrauer grouprelative curves
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The pith

Zariski-local triviality of a G-torsor over a relative curve lifts from the residue field to the base when the isogeny kernel is étale or the residue field is large.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that for a G-torsor E over a smooth projective curve C over a Henselian local ring R, if the kernel of the central isogeny G^{sc} ×_C rad(G) → G is étale over C or the residue field κ is large, then Zariski-local triviality of E over the special fiber C_κ implies Zariski-local triviality of E over C. This matters because it transfers local triviality properties from the closed fiber upward and produces new local-global statements for torsors over function fields of curves over Henselian DVRs. The authors also give an averaged version assuming only that rad(G) is isotrivial and a version for projective homogeneous spaces with no restrictions on G, all proved via a new relative and arithmetic comb smoothing technique.

Core claim

If either the kernel of the central isogeny G^{sc} ×_C rad(G) → G is étale over C or κ is large, the Zariski-local triviality of E_κ → C_κ implies the Zariski-local triviality of E → C. An averaged form holds assuming only that rad(G) is isotrivial, and a variant applies to projective homogeneous spaces under no restrictions on G.

What carries the argument

relative and arithmetic version of the comb smoothing technique applied to compactifications of torsors

If this is right

  • Local-global principle for torsors over function fields of curves over Henselian discrete valuation rings, strengthening earlier results.
  • Henselian version of the Drinfeld-Simpson theorem.
  • Injectivity result for the Brauer-Azumaya group of C.
  • Variant results for projective homogeneous spaces without restrictions on G.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The relative comb smoothing method may extend to prove similar lifting statements for other cohomology classes or bundles over relative curves.
  • The technique could connect to arithmetic versions of the Grothendieck-Serre conjecture over Henselian bases.
  • Analogous injectivity statements might hold for other invariants such as higher cohomology groups in this setting.

Load-bearing premise

C is a smooth projective curve over the Henselian local ring R with geometrically connected fibers and G is a reductive C-group with isotrivial radical torus.

What would settle it

A counterexample consisting of a reductive group G over such a C together with a G-torsor E that is Zariski locally trivial over C_κ but not over C, when the isogeny kernel is not étale and κ is not large.

read the original abstract

Let $R$ be a Henselian local ring, let $\kappa$ be the residue field of $R$, let $C$ be a smooth projective curve over $R$ with geometrically connected fibers, let $G$ be a reductive $C$-group with isotrivial radical torus $\mathrm{rad}(G)$, and let $E\to C$ be a $G$-torsor. We show that, if either the kernel of the central isogeny $G^{\mathrm{sc}}\times_C \mathrm{rad}(G)\to G$ is \'etale over $C$ or $\kappa$ is large, the Zariski-local triviality of $E_\kappa\to C_\kappa$ implies the Zariski-local triviality of $E\to C$. We also prove an averaged form of this result, assuming only that $\mathrm{rad}(G)$ is isotrivial, as well as a variant for projective homogeneous spaces under no restrictions on $G$. As consequences, we obtain a local-global principle for torsors over function fields of curves over Henselian discrete valuation rings, strengthening work of Gille--Parimala--Suresh, a Henselian version of a theorem of Drinfeld--Simpson, and an injectivity result for the Brauer--Azumaya group of $C$ not covered by earlier work of Colliot-Th\'el\`ene--Ojanguren--Parimala. Our proofs are geometric and rely on compactifications of torsors and on a relative and arithmetic version of the comb smoothing technique, which we develop in detail, building on work of Koll\'ar and Graber--Harris--Starr.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves that if E → C is a G-torsor with G a reductive group scheme over a smooth projective curve C over a Henselian local ring R (with geometrically connected fibers) and with isotrivial radical torus, then under the additional hypothesis that the kernel of the central isogeny G^{sc} ×_C rad(G) → G is étale over C or that the residue field κ is large, Zariski-local triviality of the special fiber E_κ → C_κ implies Zariski-local triviality of E → C. An averaged form of the result is established assuming only isotriviality of rad(G), together with a variant for projective homogeneous spaces with no restrictions on G. The proofs rely on compactifications of torsors and a relative/arithmetic version of comb smoothing, developed in detail from the techniques of Kollár and Graber–Harris–Starr. Consequences include a local-global principle for torsors over function fields of curves over Henselian DVRs (strengthening Gille–Parimala–Suresh), a Henselian version of the Drinfeld–Simpson theorem, and an injectivity result for the Brauer–Azumaya group of C.

Significance. If the central claims hold, the work supplies new lifting results for local triviality of torsors and homogeneous spaces over relative curves, directly strengthening the local-global principle of Gille–Parimala–Suresh and providing a Henselian analogue of Drinfeld–Simpson together with a Brauer-group injectivity statement outside the scope of Colliot-Thélène–Ojanguren–Parimala. The explicit construction of the relative arithmetic comb-smoothing technique, building on Kollár and Graber–Harris–Starr without reduction to fitted quantities or self-citations, constitutes a reusable technical contribution to the study of homogeneous spaces in arithmetic geometry.

minor comments (2)
  1. [Abstract] Abstract, final sentence: the phrase 'building on work of Kollár and Graber--Harris--Starr' would benefit from a parenthetical pointer to the specific results (e.g., the comb-smoothing statements) that are being extended; this would clarify the novelty of the relative/arithmetic adaptation already in the abstract.
  2. [§1 (Introduction)] The opening paragraph of the abstract states that rad(G) is isotrivial; the main theorem statements in the body should explicitly recall this hypothesis in the same wording to avoid any ambiguity when the averaged form (which drops the étale-kernel condition) is stated.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the report, so there are no specific points requiring point-by-point response or revision.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation develops relative comb smoothing and torsor compactifications explicitly as new geometric tools over the Henselian base, citing only external prior results of Kollár and Graber–Harris–Starr for the classical case. The central implication (Zariski-local triviality of E_κ → C_κ implies the same for E → C under the étale-kernel or large-κ hypotheses) is obtained by direct geometric arguments from the stated setup (smooth projective curve over Henselian R, reductive G with isotrivial radical torus) without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. All steps remain independent of the target conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions about curves and groups; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)
  • domain assumption C is a smooth projective curve over R with geometrically connected fibers
    Explicitly stated as part of the setup for the main theorem.
  • domain assumption G is a reductive C-group with isotrivial radical torus rad(G)
    Stated in the hypotheses of the main result.

pith-pipeline@v0.9.1-grok · 5833 in / 1251 out tokens · 26475 ms · 2026-07-03T06:00:08.840985+00:00 · methodology

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