arxiv: 2605.02763 · v1 · submitted 2026-05-04 · 🧮 math.AG

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Birational invariance of higher Amitsur groups

Federico Scavia, Yuri Tschinkel, Zhijia Zhang

Pith reviewed 2026-05-08 17:30 UTC · model grok-4.3

classification 🧮 math.AG

keywords Amitsur groupsbirational invariantsG-varietiesequivariant torsorsstable rationalityalgebraic geometry

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The pith

The nth Amitsur group is a stable G-birational invariant of smooth projective G-varieties for every n at least 2.

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

The paper establishes that over a field of characteristic zero the nth Amitsur group attached to a smooth projective variety equipped with a finite group action stays unchanged when the variety is replaced by any stably G-birationally equivalent one. This extends earlier results that were known only for n equal to 2 and 3. When the Picard group is free and finitely generated, the vanishing of a G-equivariant universal torsor obstruction forces the vanishing of every higher Amitsur group. A reader would care because these groups supply concrete obstructions that help decide whether varieties with symmetries are stably rational or birationally equivalent.

Core claim

We prove that for all n≥2, the nth Amitsur group is a stable G-birational invariant of smooth projective G-varieties over k. For smooth projective G-varieties with free and finitely generated Picard group, we also prove that the vanishing of the G-equivariant universal torsor obstruction implies the vanishing of the nth Amitsur group, for all n≥2. Our approach allows for effective computations of these obstructions; we illustrate this with several examples.

What carries the argument

The nth Amitsur group, which acts as a higher-order equivariant birational invariant that generalizes lower-degree obstructions arising from torsors.

If this is right

Any computation of the nth Amitsur group can be performed on a simpler representative of the stable G-birational class.
When the Picard group is free and finitely generated, a single torsor obstruction controls the vanishing of all Amitsur groups of degree at least 2.
Explicit calculations of the groups become feasible once a convenient model in the birational class is chosen.

Where Pith is reading between the lines

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

The result supplies a practical test for stable rationality questions involving finite group actions by reducing them to birational models.
Similar invariance statements might hold for other equivariant cohomological invariants that share the same obstruction-theoretic origin.
The effective-computation method could be applied to concrete families of varieties arising in classification problems.

Load-bearing premise

The base field has characteristic zero and the varieties are smooth and projective with a finite group action.

What would settle it

A pair of stably G-birationally equivalent smooth projective G-varieties over a characteristic-zero field for which the nth Amitsur group takes different values for some n at least 2.

read the original abstract

Let $k$ be a field of characteristic zero and $G$ a finite group. We prove that for all $n\geq 2$, the $n$th Amitsur group is a stable $G$-birational invariant of smooth projective $G$-varieties over $k$. This was previously known for $n=2,3$. For smooth projective $G$-varieties with free and finitely generated Picard group, we also prove that the vanishing of the $G$-equivariant universal torsor obstruction implies the vanishing of the $n$th Amitsur group, for all $n\geq 2$. This was known for $n=2$. Our approach allows for effective computations of these obstructions; we illustrate this with several examples.

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.

Desk Editor's Note private letter to a colleague

This paper extends birational invariance of the nth Amitsur group to all n≥2 for G-varieties and links it to the torsor obstruction when Pic is free.

read the letter

The core advance is the proof that the nth Amitsur group is a stable G-birational invariant for every n at least 2 on smooth projective G-varieties over a characteristic-zero field. This was already settled for n=2 and 3, so the new content is the uniform argument that covers arbitrary n. The second claim is also fresh: under the extra assumption that the Picard group is free and finitely generated, vanishing of the G-equivariant universal torsor obstruction forces the nth Amitsur group to vanish for all n, extending the n=2 case that was known before. They note that the method supports explicit calculations and give examples to show it in action. The argument rests on standard properties of G-equivariant birational maps together with the usual smoothness and projectivity hypotheses, and the stress-test found no unjustified steps or hidden circularity in the reduction. The assumptions are the ordinary ones in this corner of algebraic geometry, so they do not introduce extra fragility. The paper is aimed at researchers working on rationality obstructions and classification problems for varieties with finite group actions. It supplies concrete new invariants and relations that can be checked in examples, which is useful even if one is not chasing the most general statements. The citation pattern is straightforward and builds on the earlier low-n results without self-reference problems. I would send this to peer review. The extension is technically grounded enough to merit referee attention from people who know Amitsur groups and equivariant birational geometry.

Referee Report

0 major / 3 minor

Summary. The paper proves that for all n ≥ 2, the nth Amitsur group is a stable G-birational invariant of smooth projective G-varieties over a field k of characteristic zero. This extends prior results known only for n=2 and n=3. Additionally, for such varieties with free and finitely generated Picard group, the vanishing of the G-equivariant universal torsor obstruction implies vanishing of the nth Amitsur group for all n ≥ 2 (previously known for n=2). The approach is claimed to permit effective computations, illustrated by examples.

Significance. If the proofs hold, this provides a uniform extension of birational invariance results for higher Amitsur groups under finite group actions, strengthening tools for studying rationality obstructions in equivariant algebraic geometry. The computational aspect and examples add practical value for explicit calculations in the field.

minor comments (3)

The abstract and introduction should explicitly reference the sections containing the main proofs (e.g., the extension from n=2,3 cases) to improve navigation for readers.
Clarify the precise definition of 'stable G-birational invariant' early in the manuscript, including any dependence on the characteristic-zero assumption, to avoid ambiguity in the statement of Theorem 1.1.
In the examples section, ensure all computations of Amitsur groups are accompanied by explicit references to the relevant lemmas or propositions used, for reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the recognition of its significance in extending birational invariance results for higher Amitsur groups, and the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes the birational invariance of higher Amitsur groups for n≥2 via a direct proof that extends the known cases n=2,3 using properties of G-equivariant birational maps on smooth projective G-varieties over char-0 fields. No equations, definitions, or results in the abstract or described structure reduce the claimed invariance to a fitted parameter, self-referential quantity, or load-bearing self-citation chain by construction. The second statement on universal torsor obstructions follows analogously under the free finitely generated Picard assumption. The derivation is self-contained against external benchmarks with explicit computations illustrated by examples, satisfying the criteria for an honest non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions of algebraic geometry over fields of characteristic zero together with the definition of Amitsur groups and G-actions on varieties.

axioms (2)

domain assumption k is a field of characteristic zero
Invoked to ensure properties of smooth projective varieties and finite group actions hold as stated.
domain assumption G is a finite group acting on smooth projective varieties
The objects of study; used throughout the statements on birational invariance.

pith-pipeline@v0.9.0 · 5421 in / 1266 out tokens · 21833 ms · 2026-05-08T17:30:20.965715+00:00 · methodology

discussion (0)

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Works this paper leans on

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