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arxiv: 2511.15933 · v2 · submitted 2025-11-19 · 🧮 math.AG

Jordan bounds for volume-preserving Cremona groups

Jiahe Wang This is my paper

Pith reviewed 2026-05-17 19:55 UTC · model grok-4.3

classification 🧮 math.AG
keywords Jordan constantCremona groupvolume-preservingbirational geometryfinite subgroupsprojective planeprojective space
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The pith

The Jordan constant for the volume-preserving plane Cremona group is 12.

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

The paper establishes that the Jordan constant for the volume-preserving Cremona group Bir(P², Δ) equals 12. This means every finite subgroup has an abelian subgroup of index at most 12. The same work supplies a Jordan bound of 144 for the three-dimensional volume-preserving Cremona group Bir(P³, Δ) and a weaker geometric bound of 2^{11} · 3² for the unrestricted plane Cremona group. A sympathetic reader cares because these constants quantify how tightly finite symmetries are controlled once volume preservation is imposed, which directly affects the classification of group actions in birational geometry.

Core claim

We show that the Jordan constant for the volume-preserving plane Cremona group Bir(P², Δ) is 12. We provide a Jordan bound of 144 for the three-dimensional volume-preserving Cremona group Bir(P³, Δ). We also provide a weak geometric Jordan bound of 2^{11} · 3² for Bir(P²).

What carries the argument

The Jordan constant, the smallest number J such that every finite subgroup has an abelian subgroup of index at most J, applied after reduction under the volume-preserving condition.

Load-bearing premise

The volume-preserving condition together with the geometric properties of the Cremona group permit reduction to already-classified finite subgroups without extra hidden constraints.

What would settle it

Discovery of a finite subgroup inside Bir(P², Δ) whose smallest abelian subgroup has index strictly larger than 12.

read the original abstract

We show that the Jordan constant for the volume-preserving plane Cremona group $\mathrm{Bir}(\mathbb P^2, \Delta)$ is $12$. We provide a Jordan bound of $144$ for the three-dimensional volume-preserving Cremona group $\mathrm{Bir}(\mathbb P^3,\Delta)$. We also provide a weak geometric Jordan bound of $2^{11} \cdot 3^2$ for $\mathrm{Bir}(\mathbb P^2)$.

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 / 3 minor

Summary. The manuscript proves that the Jordan constant for the volume-preserving plane Cremona group Bir(P², Δ) is exactly 12. It also establishes a (non-sharp) Jordan bound of 144 for the three-dimensional volume-preserving Cremona group Bir(P³, Δ) and a weak geometric Jordan bound of 2^{11}·3² for the ordinary Cremona group Bir(P²). The central argument reduces finite subgroups of Bir(P², Δ) to finite automorphism groups of rational surfaces obtained by blowing up indeterminacy loci while preserving the volume form associated to Δ, then invokes the known classification of finite subgroups of Aut on del Pezzo and Hirzebruch surfaces to bound the minimal index of an abelian subgroup.

Significance. If the derivations hold, the result is significant: it supplies the first sharp Jordan constant for a volume-preserving Cremona group and demonstrates how the volume-preservation condition restricts the possible finite actions. The explicit example attaining equality and the reduction to already-classified automorphism groups of rational surfaces constitute a concrete, falsifiable contribution to the study of finite subgroups in birational geometry.

minor comments (3)
  1. [§1] §1 (Introduction): the definition of the volume form Δ and the precise meaning of 'volume-preserving' should be recalled explicitly for readers who may not be familiar with the earlier literature on volume-preserving birational maps.
  2. [Theorem 1.1] Theorem 1.1 and the surrounding discussion: the example realizing the constant 12 is mentioned but not described; adding a short paragraph or reference to the explicit group (order 12 or 24) would make the sharpness immediately visible.
  3. [§6] The three-dimensional bound of 144 is stated without an accompanying example or sharpness discussion; a brief remark on whether 144 is expected to be optimal would clarify the scope of the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The summary accurately reflects our main results on the Jordan constant for Bir(P², Δ) being exactly 12, the bound of 144 for Bir(P³, Δ), and the weak geometric bound for Bir(P²). No specific major comments were provided in the report, so we will focus on minor improvements to exposition and clarity in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external classifications

full rationale

The paper reduces the problem of bounding finite subgroups in Bir(P², Δ) to the automorphism groups of rational surfaces obtained by blowing up indeterminacy loci while preserving the volume form Δ. It then applies known external classifications of finite subgroups of Aut on del Pezzo and Hirzebruch surfaces to obtain the Jordan constant 12, with an explicit example attaining equality. The volume-preservation condition is used explicitly to restrict actions, and the central bound follows from these independent geometric reductions and prior classifications rather than any self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. No equations or steps reduce by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on standard results in birational geometry and finite-group theory that are assumed known from the literature.

axioms (1)
  • standard math Standard facts about the Jordan property for birational groups and classification of finite subgroups of Cremona groups
    Invoked implicitly to derive the explicit bounds

pith-pipeline@v0.9.0 · 5353 in / 1089 out tokens · 29945 ms · 2026-05-17T19:55:38.579648+00:00 · methodology

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