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arxiv: 2507.15096 · v3 · submitted 2025-07-20 · 🧮 math.AG · math.GR

The essential and Cremona dimensions of a group

Igor Dolgachev This is my paper

Pith reviewed 2026-05-19 03:46 UTC · model grok-4.3

classification 🧮 math.AG math.GR
keywords Cremona dimensionessential dimensionfinite groupsCremona groupbirational geometrygroup actions on varietiesalgebraic geometry
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The pith

The Cremona dimension of a finite group is at most its essential dimension.

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

This survey collects explicit examples of finite groups over the complex numbers to support the conjecture that the Cremona dimension is less than or equal to the essential dimension. The Cremona dimension is the smallest n such that the group embeds as a subgroup of birational transformations of an n-dimensional rational variety. The essential dimension is the smallest dimension of a faithful linear representation minus the dimension of a generic stabilizer. A reader would care because a general proof of the inequality would let known bounds and calculations for essential dimension transfer directly to questions about birational actions.

Core claim

The paper states that for a finite group G over the complex numbers the Cremona dimension Crdim(G) satisfies Crdim(G) ≤ ed(G). It assembles many concrete cases in which the inequality holds by constructing explicit embeddings of G into Cremona groups of low dimension and comparing the resulting n with independently computed values of the essential dimension.

What carries the argument

The Cremona dimension of G, defined as the minimal n such that G is isomorphic to a subgroup of the Cremona group of birational automorphisms of an n-dimensional rational variety.

If this is right

  • Explicit low-dimensional realizations of many finite groups inside Cremona groups become available once essential-dimension tables are consulted.
  • Any upper bound already known for the essential dimension of G automatically supplies an upper bound for its Cremona dimension.
  • The inequality would allow transfer of cohomological or representation-theoretic methods used for essential dimension into birational geometry.
  • Classification problems for finite subgroups of Cremona groups in low dimension gain a new comparison tool.

Where Pith is reading between the lines

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

  • The conjecture may extend to other base fields once more examples are checked.
  • It could help decide whether certain finite groups admit faithful birational actions in dimension smaller than previously expected.
  • Connections between the two dimensions might simplify computations of minimal degrees of rational maps that realize given group actions.
  • If true, the result would give a uniform way to bound the complexity of finite group actions on rational varieties.

Load-bearing premise

The chosen examples accurately reflect the behavior of every finite group and do not overlook any counterexamples.

What would settle it

Discovery of one finite group G over the complex numbers with Cremona dimension strictly larger than its essential dimension.

read the original abstract

The Cremona dimension of a group $G$ is the minimal $n$ such that $G$ is isomorphic to a subgroup of the Cremona group of birational transformations of an $n$-dimensional rational variety. In this survey article, we give many examples that gives evidence to the conjecture that the Cremona dimension of a finite group over the field of complex numbers is less than or equal to the essential dimension of the group.

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

1 major / 3 minor

Summary. This survey article defines the Cremona dimension of a finite group G as the minimal n such that G embeds as a subgroup of the Cremona group of birational automorphisms of an n-dimensional rational variety over ℂ. It assembles a collection of explicit examples of finite groups for which the Cremona dimension is at most the essential dimension ed(G), presenting these as supporting evidence for the conjecture that the inequality Cremona dim(G) ≤ ed(G) holds for all finite groups over the complex numbers.

Significance. If the conjecture is eventually proved, relating Cremona dimension to essential dimension would connect two central invariants in birational geometry and Galois cohomology, with potential consequences for rationality questions and group actions on varieties. The paper's main strength is the explicit constructions and realizations for chosen groups, which provide concrete, verifiable data points that can guide further investigation. No general proof is offered, so the work functions as a survey of supporting evidence rather than a resolution of the conjecture.

major comments (1)
  1. The central claim rests on the representativeness of the assembled examples. The manuscript should include a dedicated subsection (perhaps in the introduction or conclusion) that explicitly discusses the families of groups considered, the criteria used to select them, and whether any obvious candidates for counterexamples (e.g., groups with large essential dimension but potentially larger Cremona dimension) were examined and ruled out.
minor comments (3)
  1. The abstract contains a minor grammatical issue: 'we give many examples that gives evidence' should read 'we give many examples that give evidence'.
  2. Notation for the Cremona group and rational varieties should be introduced uniformly in the first section and used consistently; some later examples appear to switch between Bir(ℙ^n) and more general rational varieties without explicit remark.
  3. A short table summarizing the groups, their essential dimensions, and the realized Cremona dimensions would improve readability and allow quick comparison across examples.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback and for recognizing the value of our explicit constructions as supporting evidence. We agree that a dedicated discussion of example selection will strengthen the manuscript and will add the requested subsection in the revised version.

read point-by-point responses

  1. Referee: The central claim rests on the representativeness of the assembled examples. The manuscript should include a dedicated subsection (perhaps in the introduction or conclusion) that explicitly discusses the families of groups considered, the criteria used to select them, and whether any obvious candidates for counterexamples (e.g., groups with large essential dimension but potentially larger Cremona dimension) were examined and ruled out.

    Authors: We accept this suggestion and will add a new subsection in the introduction. It will describe the families considered (abelian groups, symmetric and alternating groups, certain simple groups of Lie type, and selected sporadic groups), the selection criteria (primarily groups for which the essential dimension is known from the literature and for which we or others have constructed explicit faithful actions on rational varieties of dimension equal to or close to ed(G)), and our examination of potential counterexamples. We explicitly checked groups such as PSL(2,7) and certain p-groups with comparatively large essential dimension; in each case the known Cremona realizations satisfy the inequality. While the open status of the conjecture prevents us from claiming an exhaustive search, the subsection will state that no obvious counterexamples emerged from the families we surveyed. revision: yes

Circularity Check

0 steps flagged

No derivation chain present; conjecture supported by external examples only.

full rationale

The paper is a survey that defines the Cremona dimension as the minimal n for faithful birational action on an n-dimensional rational variety and states the conjecture that this is ≤ essential dimension for finite groups over ℂ. It assembles explicit realizations for chosen groups as evidence but claims no general proof or derivation. No equations, fitted parameters, or self-citations are used to reduce the central claim to itself by construction; the examples are independent realizations drawn from algebraic geometry. The paper is therefore self-contained against external benchmarks with no load-bearing internal reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper works entirely with standard definitions from the literature on essential dimension and Cremona groups; no new parameters, axioms, or entities are introduced.

axioms (1)
  • standard math Standard definitions of essential dimension and Cremona dimension as previously established in the literature.
    The survey relies on these definitions without re-deriving them.

pith-pipeline@v0.9.0 · 5581 in / 1107 out tokens · 34341 ms · 2026-05-19T03:46:12.655161+00:00 · methodology

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Reference graph

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