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Representations of finite subgroups of Cremona groups

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Pith's one-line read The plane Cremona group's finite subgroups need exactly 6 dimensions over most characteristic-0 fields, 8 if √−3 is present, and no finite bound in positive characteristic.

desk verdict A genuinely valuable paper on a new invariant, with a real missing case in the main upper bound for c2(k) that a referee should catch. read the letter →

arxiv 2507.04474 v1 pith:GQ5GY3PE submitted 2025-07-06 math.AG math.GR

Alexander Duncan , Bailey Heath , Christian Urech This is my paper · ORCID
classification math.AGmath.GR MSC 14E0720C99
keywords Cremonagroupsrepresentationdimensionfinitesubgroupsbirationalautomorphismsalgebraictoriminimalrationalsurfacespositivecharacteristicprojectivelinear
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks how many dimensions are needed to faithfully represent every finite subgroup of the Cremona group—the birational automorphisms of projective n-space—over a given field, and answers this exactly for n=1 and n=2 over every field. The plane Cremona group Cr₂(k) behaves differently depending on the field: over any characteristic-0 field that does not contain √−3, six dimensions suffice and are necessary; over fields containing √−3, eight dimensions are needed and suffice; and over fields of positive characteristic, no finite bound exists. For higher n, the paper proves finiteness for many characteristic-0 fields, including number fields and the complex numbers, and gives lower bounds that grow exponentially in n. The exact values come from a case analysis of rational surfaces with group actions, combined with new bounds on the representation dimensions of automorphism groups of tori and projective spaces.

What carries the argument

The load-bearing structure is the reduction of finite subgroups of Cr₂(k) to automorphism groups of minimal rational surfaces—del Pezzo surfaces and conic bundles—following Manin and Iskovskikh. Around that hub, the argument uses three further pieces: (1) representations of algebraic tori, where certain finite groups are realized as subgroups of Aut(T) for a rational torus T built from a root lattice, turning a combinatorial 'symmetric rank' computation into lower bounds on representation dimension; (2) the projective linear groups PGL₃(k) and PGL₄(k), where the Weil representation of (C_p)² ⋊ SL₂(F_p) gives the sharp lower bound p²−1 and a Blichfeldt classification of primitive finite subgroups gives the upper bound when p=3 and √−3 ∉ k; and (3) Mal'cev's compactness theorem, which lets the positive-characteristic result be proved by showing the group k[x]⋊k is not linear over any field.

What would settle it

Run the orbit enumeration for the G₂ root lattice with its Weyl group acting on L/4L: if any W-stable subset of L/4L of size at most 5 generates the module, then the claimed lower bound c₂(k) ≥ 6 fails; the same check applies to A₃, F₄, D₅, and E₆ for the values 12, 24, 40, and 72.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is the exact value of c₂(k), the supremum of minimal faithful-representation dimensions over finite subgroups of the plane Cremona group: it is 6 for any field of characteristic 0 that does not contain √−3, 8 for characteristic-0 fields that do, and infinite in positive characteristic. The lower bound of 6 is obtained by constructing, from root lattices of types A₁, G₂, A₃, F₄, D₅, E₆ and B_n, rational tori whose automorphism groups contain finite groups with large representation dimension. The upper bound of 6 or 8 uses the Manin–Iskovskikh reduction: any finite subgroup of Cr₂(k) embeds in the automorphism group of a minimal rational surface, either a del Pezzo surface or a conic bundle surface, and the paper bounds the representation dimension of each such automorphism group in turn. The remaining gap between 6 and 8 is exactly whether k contains √−3, which controls whether PGL₃(k) contains the Hessian group (C₃)² ⋊ SL₂(F₃).

Load-bearing premise

The 6-dimensional lower bound for c₂(k) rests on an unshipped Sage computation asserting that, for the root lattices G₂, A₃, F₄, D₅, and E₆, no union of orbits of size less than the claimed minimum generates L/4L (Theorem 5.4).

Editorial extensions

If this is right

  • For every field k of characteristic 0, finite subgroups of the plane Cremona group are uniformly representable in at most 8 dimensions, and exactly 6 unless √−3 ∈ k.
  • In positive characteristic, there are finite subgroups of Cr_n(k) for every n ≥ 2 with arbitrarily large minimal faithful-representation dimension, so no finite uniform bound exists.
  • The three-dimensional complex Cremona group satisfies 15 ≤ c₃(C) ≤ 62208, giving the first nontrivial bounds of their kind for rank 3.
  • For characteristic-0 fields containing all roots of unity, c_n(k) is finite for every n, and the same holds for fields finitely generated over Q.
  • The lower bounds c_n(k) ≥ 2ⁿ for n ≥ 7 show that, on a fixed field, the minimal faithful-representation dimension of finite subgroups grows at least exponentially in the rank of the Cremona group.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The exponential lower bound 2ⁿ suggests the true growth of c_n(k) may be much faster, possibly driven by symmetric ranks of Weyl groups; whether c_n(C) is finite for all n and how it grows remains open, since the paper's finiteness proof is non-constructive.
  • The positive-characteristic proof shows k[x]⋊k is not linear over any field, a fact that may extend to other elementary birational groups and suggests finite subgroups of Cremona groups over finite fields still have unbounded representation dimension over the same field.
  • The unshipped Sage computation that certifies the lower bounds is a testable artifact: re-running the orbit enumeration for the G₂, A₃, F₄, D₅, and E₆ root lattices would independently confirm or correct the 6-, 12-, 24-, 40-, and 72-dimensional lower bounds.
  • The same torus-and-root-lattice method likely applies to other groups of birational transformations, such as automorphism groups of rational varieties with torus actions, where symmetric rank may again control representation dimension.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper defines c_n(k) as the supremum over finite subgroups G of Cr_n(k) of the minimal dimension of a faithful linear representation of G over k. It determines c_1(k) for all fields, determines c_2(k) explicitly (∞ in positive characteristic, 8 if √−3∈k, 6 otherwise), proves that c_n(k)=∞ for n≥2 in positive characteristic, proves finiteness of c_n(k) for many characteristic-0 fields, and gives exponential lower bounds in n. The proofs use minimal rational G-surfaces, algebraic tori and their symmetric ranks, weighted projective spaces, and a GAP-checked group-theoretic appendix.

Significance. The exact evaluation of c_2(k) is a major step in understanding finite subgroups of plane Cremona groups over arbitrary fields, and the lower-bound technology via symmetric ranks of root lattices is original and useful. The positive-characteristic non-linearity result for Cr_n(k) (n≥2) and the use of model-theoretic compactness are elegant. The appendix's hand proof of the six-dimensional bounds for extensions of polyhedral groups, accompanied by GAP verification, is a valuable resource. If the gaps identified below are repaired, the paper will be a strong contribution.

major comments (3)
  1. [§8, Proposition 8.3 and proof of Theorem 1.2] Proposition 8.3 claims fdim_k(Aut(X)) ≤ 6 for every rational del Pezzo surface X not isomorphic to P^2. The proof handles degrees 8 and 7 by equivariant blowdown, then states that the remaining cases of degree d ≤ 6 follow from Corollary 7.5. However, Corollary 7.5 gives fdim_k(Aut(X)) ≤ d+1 for d ≥ 3, so for d=6 it yields the bound 7, not 6. Degree-6 del Pezzo surfaces are rational and can be k-forms over fields without √−3, precisely in the regime where Theorem 1.2 asserts c_2(k)=6; moreover Theorem 8.1 permits a minimal G-surface of this type with Pic(X)^G ≅ Z. The upper bound c_2(k) ≤ 6 is therefore not proven as written. A separate argument for the degree-6 del Pezzo case (e.g., using the structure of its anticanonical ring) is required.
  2. [§3, proof of Theorem 1.1, characteristic-2 descent] In the case where ρ is not absolutely irreducible and b=0 for every g∈G, the proof asserts that since ρ is defined over L∩K and finite fields are perfect, 'K = k', and hence ρ is defined over k. This inference is invalid: K/k may be a non-trivial purely inseparable extension while K contains a finite subfield such as L∩K. The equality K=k does not follow, and the descent of ρ to a representation over k is not established. Thus the upper bound c_1(k) ≤ 2 for infinite imperfect fields of characteristic 2 rests on an unsupported step. Please supply a correct descent argument or an alternative proof for this case.
  3. [§5, proof of Theorem 5.4] The values symrank(L/dL,W)=|Ω| for n=1,...,6 in Table 1 are justified by the sentence 'This was checked using [Sage]', with no code or output included. This finite verification is load-bearing: it underpins the lower bounds in Theorem 1.5 (including c_2(k) ≥ 6 in Theorem 1.2). Please provide the Sage script and its output, or give a short hand-checkable certificate (e.g., listing the orbit sizes and a generating orbit union for each lattice).
minor comments (5)
  1. [§3, proof of Theorem 1.1] The term 'relative closure of the prime field F_2 in F' is not defined; please make explicit that it is the algebraic closure of F_2 inside F (a finite field).
  2. [Table 2, Appendix A] The notations gS4+ and gS4− for the binary octahedral groups are used without definition; please define them and explain the GAP ID column.
  3. [§4, Lemma 4.5] The condition (b) in the proof appears garbled: the displayed expression with 'W_i' and 'W_{i≠j}' should be a disjunction expressing that the matrix (x^g_{ij}) is not the identity. Please correct the notation.
  4. [§6, Proposition 6.4] In the reducible case, the claim that the representation σ∨⊗τ factors through eG→G and is faithful is stated without justification; adding one sentence on why the kernel is exactly the central scalars would help.
  5. [§8, Proposition 8.6] The assertion that a rational k-form of P(1,1,n,n) is isomorphic to the standard weighted projective space is not immediate; please include a reference or a short justification.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular step found; the main concerns are a non-circular degree-6 del Pezzo gap in the proof of Theorem 1.2 and undocumented Sage/GAP finite checks.

full rationale

The paper's central derivations are direct and do not reduce by construction to their own inputs. c2(k) is defined independently as a supremum of representation dimensions, and the lower bounds are obtained by explicitly constructed tori whose finite subgroups provably require faithful representations of the stated degree (Lemma 5.3 and Theorem 5.4). The symmetric-rank values in Table 1 are supported by a stated finite enumeration in Sage plus an analytic argument for n ≥ 7; the citation to [Hea24] supplies the same choices of Ω and L but does not replace the verification. The intro's citation of [Ure21] for c2(C) ≤ 48 is contextual, not an input to Theorem 1.2. The appeal to [BCDP22] in Lemma 7.3 concerns a standard linearization statement in a separate published paper, so it is independent support rather than a circular self-citation. Two incompletenesses are not circularity but should be flagged. First, Proposition 8.3 states that remaining del Pezzo cases 'are of degree d ≤ 6, which follow from Corollary 7.5,' yet Corollary 7.5 gives d + 1 for d ≥ 3; for d = 6 this yields only 7, not 6. Degree-6 rational del Pezzo k-forms are not otherwise handled in Section 8, so the upper bound c2(k) ≤ 6 in Theorem 1.2 has a missing case. Second, the finite checks in Theorem 5.4 ('This was checked using [Sage]') and in Appendix A (GAP Small Groups Database) are not shipped, so those lower bounds rest on undocumented computation. Neither issue is a case of a claimed result being equivalent to its inputs; the derivation chain is otherwise self-contained, and the self-citations present are not load-bearing in a circular sense.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

This is a pure mathematics paper. It defines no new entities and fits no parameters. The central claims rest on a set of deep published theorems (Manin-Iskovskikh, Birkar, Prokhorov-Shramov, Blichfeldt, Shephard-Todd, Beauville, Mal'cev compactness, Hilbert 90, Serre trace-field descent) and on finite computer checks that are described but not shipped.

assumptions (9)
  • standard math Manin-Iskovskikh classification of minimal rational G-surfaces over arbitrary fields of characteristic 0
    Theorem 8.1 reduces finite subgroups of Cr2(k) to actions on del Pezzo surfaces or conic bundles; the paper cites [DI09b].
  • standard math Birkar boundedness of Fano varieties and Prokhorov-Shramov Jordan property for Cremona groups
    Used to prove finiteness of c_n(k) in Theorem 1.4 and the upper bound in Theorem 9.2; cited from [Bir21] and [PS16].
  • standard math Blichfeldt classification of primitive finite subgroups of PGL3(C)
    Used in Proposition 6.4 to bound fdim_k(PGL3(k)); cited from [Bli17].
  • standard math Beauville classification of finite subgroups of PGL2(K)
    Used in Theorem 1.1 and Appendix A; cited from [Bea10].
  • standard math Shephard-Todd classification and Benard table for reflection group G29
    Used in Proposition 6.5 to show fdim(PGL4(k)) = 15; cited from [Ben76].
  • standard math Compactness theorem for first-order logic (Mal'cev)
    Used in Lemma 4.5 to prove non-linearity of k[x] ⋊ k; cited from [Mal40].
  • standard math Hilbert's Theorem 90
    Used in Lemma 2.2 for invariance under k-forms; cited from [Ser02].
  • standard math Serre's trace-field descent for finite group representations
    Used in Theorem 1.4 to descend representations to fields containing all roots of unity; cited from [Ser77].
  • standard math Cox and Liendo-Lucchini Arteche description of automorphisms of toric varieties over arbitrary fields
    Used in Lemma 7.1 for weighted projective space automorphism groups; cited from [Cox95], [PS17], [LLA22].

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Pith. "Pith review of Representations of finite subgroups of Cremona groups." pith.science (2026). https://pith.science/paper/GQ5GY3PE

@misc{pith2026250704474,
  author       = {Pith},
  title        = {Pith review of: Representations of finite subgroups of Cremona groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GQ5GY3PE}},
  note         = {Machine review of arXiv:2507.04474}
}
read the original abstract

The Cremona group of rank n over a field k is the group of birational automorphisms of the n-dimensional projective space over the field k. We study the minimal dimension such that all finite subgroups of the Cremona group have a faithful representation of that dimension over the same field. We find the exact value for rank 1 and 2 over all fields. We prove that the value is infinite for all fields of positive characteristic and rank greater than one. For many fields of characteristic 0, which include number fields and the complex field, we show that the value is finite for all ranks. Finally, for all fields of characteristic 0, we prove that the dimension is bounded below by a function that is exponential in the rank.

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