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Linearization problem for finite subgroups of the plane Cremona group

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Pith's one-line read A finite subgroup of the plane Cremona group is linearizable exactly when its minimal model appears on a short explicit list.

desk verdict A complete-looking solution to the linearization problem for finite subgroups of Cr2(k); the main theorem is very likely correct, but two key converse arguments are only implicit. read the letter →

arxiv 2412.12022 v1 pith:AGULDHTY submitted 2024-12-16 math.AG math.GR

Antoine Pinardin , Arman Sarikyan , Egor Yasinsky This is my paper · ORCID
classification math.AGmath.GR MSC 14E0714E0514E3014J4514M22
keywords planeCremonagrouplinearizationG-MorifibrespaceSarkisovprogramdelPezzosurfacesHirzebruchfiniteactionsbirationalgeometry
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 claims a complete solution of the linearization problem for finite subgroups of the plane Cremona group over an algebraically closed field of characteristic zero. It asserts that after regularizing a finite subgroup $G \subset \mathrm{Cr}_2(k)$ on a two-dimensional $G$-Mori fibre space, $G$ is linearizable — conjugate in the Cremona group to a subgroup of $\mathrm{PGL}_3(k)$ — if and only if that surface with its $G$-action appears on a short explicit list. The list covers $G$-conic bundles with $K^2=8$ satisfying parity conditions, and $G$-del Pezzo surfaces of degrees $5,6,8,9$ with specified cyclic, dihedral, or symmetric groups. If the theorem is correct, the long-standing classification question reduces to a membership test.

What carries the argument

The machinery is the equivariant Sarkisov program in dimension 2: every $G$-birational map between $G$-Mori fibre spaces decomposes into Sarkisov $G$-links of types I, II, III, and IV, each a blow-up followed by a contraction. The paper combines this with orbit-length arithmetic on $\mathbb{P}^1$ (Klein's classification) and on $\mathbb{P}^1 \times \mathbb{P}^1$ (Goursat fibre products) to decide which chains of links can reach $\mathbb{P}^2$, and with explicit $G$-elementary transformations on Hirzebruch surfaces to construct linearizations when possible. The Euclidean algorithm in Proposition 6.19 is the constructive core for the dihedral fibre-product cases.

What would settle it

Test the converse of Theorem 6.1 directly: take a $G$-conic bundle $F_n$ with $n$ even whose base action has orbit lengths with greatest common divisor 1 but is neither cyclic nor dihedral of odd order (for instance a base image isomorphic to $A_4$), and search for a chain of $G$-elementary transformations to $F_1$; the theorem predicts none exists, so producing one would refute the Main Theorem.

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Extended reading notes

Core claim

The central claim is the Main Theorem. Let $k$ be algebraically closed of characteristic zero and let $G \subset \mathrm{Cr}_2(k)$ be finite. Choose a regularization of $G$ on a two-dimensional $G$-Mori fibre space $S$ over the base $B$. Then $G$ is linearizable if and only if $(S,G)$ is one of the following: a $G$-conic bundle over $B \simeq \mathbb{P}^1$ with $K_S^2 = 8$, namely a Hirzebruch surface $F_n$ with $n$ odd (any $G$), or $F_n$ with $n>0$ even where $G$ acts on $B$ cyclically or as $D_{2m+1}$, or the quadric $F_0 \simeq \mathbb{P}^1 \times \mathbb{P}^1$ with $G$ equal to $C_n \times_Q C_m$, $C_n \times_Q D_{2m+1}$, or a dihedral $D_{2n+1} \times_Q D_{2m+1}$; or a $G$-del Pezzo surface, namely the quintic with $C_5$ or $D_5$, the sextic with $C_6$ or $S_3$, the quadric $\mathbb{P}^1 \times \mathbb{P}^1$ with $(C_n \times_Q C_n)\bullet C_2$, or $\mathbb{P}^2$ with Blichfeldt's list. Here $\times_Q$ denotes a fibre product over a common quotient and $\bullet$ denotes an extension. The list is both necessary and sufficient, so every finite subgroup not represented by one of these models is non-linearizable.

Load-bearing premise

The load-bearing premise is that the classification of equivariant Sarkisov links in dimension 2 is exhaustive, so that every $G$-birational map from a $G$-Mori fibre space to $\mathbb{P}^2$ decomposes into links of types I, II, III, and IV.

Editorial extensions

If this is right

  • The linearization problem for the plane Cremona group is completely solved in the stated setting.
  • Every finite subgroup whose minimal model is a $G$-conic bundle with $K^2 \in \{1,2,4\}$ is non-linearizable; the only linearizable conic bundles in the list have $K^2 = 8$.
  • On del Pezzo surfaces of degrees 5 and 6, only the listed cyclic and small symmetric or dihedral groups are linearizable; the $D_6$-action on the sextic is not linearizable.
  • Linearizability can be decided by computing a regularization, running the $G$-minimal model program, and comparing the resulting model with the list.
  • Stable linearizability is genuinely weaker than linearizability in dimension 2: the $D_6$-action on the sextic is stably linearizable but not linearizable.

Reading between the lines

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

  • One testable extension is to automate the criterion: given generators of a finite subgroup of $\mathrm{Cr}_2(k)$, a computer algebra system could compute the $G$-Mori fibre space and check membership in the list; the paper does not provide such an implementation.
  • The Euclidean-algorithm step used for dihedral fibre products is constructive, so the birational maps it produces could be assembled into explicit conjugating transformations for the linearizable cases.
  • Over non-closed fields such as $\mathbb{R}$ or $\mathbb{Q}$, the list is not expected to survive unchanged: the orbit-counting arguments rely on algebraic closure, and fixed-curve rigidity is already known to behave differently over $\mathbb{R}$.
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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

2 major / 4 minor

Summary. The paper claims a complete classification of linearizable finite subgroups of the plane Cremona group over an algebraically closed field of characteristic zero. The Main Theorem states that a finite group G ⊂ Cr2(k) is linearizable if and only if a regularization on a G-Mori fibre space is one of the following: a G-conic bundle over P1 of a specified Hirzebruch-surface or quadric type, or a G-del Pezzo surface of degree 5, 6, 8, or 9 with the specified group actions (including Blichfeldt's list for P2). The proof uses the equivariant Sarkisov program, reduces to G-del Pezzo surfaces of degree at least 4 and to conic bundles with no singular fibres, and then analyzes each case. The paper also contains a self-contained classification of finite subgroups of Aut(P1 × P1) with supporting Magma code.

Significance. If the Main Theorem is fully established, this is a major result: it solves a long-standing open problem and provides a complete, explicit list of linearizable finite subgroups of Cr2(k). The systematic use of the equivariant Sarkisov program, together with the detailed group-theoretic case analysis, is appropriate and the paper is clearly a substantial contribution to the field. The authors provide reproducible Magma code for the quadric-automorphism classification, and the overall strategy of reduction to K2 ≥ 4 del Pezzo surfaces and to no-singular-fibre conic bundles is sound. The main caveats are two places where the text states or relies on load-bearing implications without giving the full verification: the converse direction of Theorem 6.1 and the 'crucial observation' in Section 6.3. These are fixable but are necessary for the claimed completeness.

major comments (2)
  1. [§6.1, Theorem 6.1] Theorem 6.1 is stated as an if-and-only-if, but the proof in §6.1 establishes only the sufficiency direction: Corollary 6.8 covers cyclic and odd-dihedral base actions, and Corollary 6.9 covers odd n. The necessity direction — that for even n with \hat G neither cyclic nor isomorphic to D_{2m+1}, the G-conic bundle F_n is not G-birational to F1 or P2 — is never written. This is load-bearing for the first two rows of the Main Theorem. Please add a proof of this necessity, or state and justify a reduction to the rank-2 quadric classification (Theorem 6.14) and explain how the hypotheses of that theorem apply.
  2. [§6.3, 'crucial observation'] The paragraph after Proposition 5.2 in Section 6.3 asserts that for a rank-2 quadric S = F0, G is linearizable if and only if there is a sequence of G-elementary transformations and type IV links from F0 to F1 followed by a type III contraction, and that this follows from [DI09, Propositions 7.12, 7.13]. No derivation is given. This observation is the starting point of Theorem 6.14 and therefore underlies the rank-2 quadric rows of the Main Theorem. Please provide the missing verification, in particular explaining why no other Sarkisov link types (e.g., type I or type II links centred at orbits of length 1, 2, 3, or 5) can occur or are already accounted for in the stated chain.
minor comments (4)
  1. [§6.2, Proposition 6.13] In the case |Σ| = 2, the sentence 'the points E2 ∩ E3 and E5 ∩ E6 are unique G-fixed points on T' and the later conclusion about T' are somewhat compressed; a short explanation of why no other G-fixed points appear after elementary transformations would improve readability.
  2. [§6.2, Lemma 6.17] The phrase 'the lengths of the orbits is preserved under G-fibrewise transformations' is not literally true for orbits on the total space; what is preserved is the length of the induced orbit on the base P1. The parity argument in the lemma is convincing once this is clarified, but the wording should be corrected.
  3. [Main Theorem, table] In the table row for the quadric F0, the entry 'D_{2n+1} ×Q D_{2m+1} is dihedral' uses n,m without explicitly stating the range; Section 6.3 uses n,m ≥ 3, but the table would benefit from restating this condition.
  4. [Section 5, Theorem 5.6] The table entries labelled 'No id' could confuse readers; it would be helpful to state explicitly that these families are infinite and therefore have no single GAP ID.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the derivation is largely self-contained; the few overlapping-author citations are imported auxiliary lemmas, not the target conclusion.

full rationale

The Main Theorem is derived from the equivariant Sarkisov program rather than from its own statement. Sufficiency directions are constructive, using explicit G-birational maps to P2 or F1 (Corollaries 6.8 and 6.9, Proposition 6.19), and necessity directions are exclusion arguments based on orbit-length arithmetic and the external classification of Sarkisov links due to Iskovskikh and Dolgachev--Iskovskikh ([Isk96], [DI09]). No parameter is fitted, no prediction is defined in terms of the target, and the linearizability list is not used as an input to itself. The only overlapping-author citations are Lemma 4.14 from [Yas24] and Lemma 4.16 from [Pin24b]; these describe automorphism-group structures of degree-6 del Pezzo surfaces, which is not the linearization statement, and Lemma 4.16 is accompanied by a direct proof in the text. They support the degree-6 row of the Main Theorem but do not by themselves establish it, so this is legitimate importation of prior work rather than circularity. The paper does contain gaps that a referee should weigh as correctness risks: Section 6.3's 'crucial observation' is asserted to follow from [DI09, Propositions 7.12, 7.13] without verification, and the necessity direction of Theorem 6.1 is not written out, requiring an implicit reduction to F0 and the rank-2 classification. These are omissions or reliance on external completeness, not circular reductions: the cited classification is external, and the omitted converse does not assume the Main Theorem. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper imports several major classifications as black boxes: finite subgroups of PGL2 and PGL3, the equivariant Sarkisov program and link classification, Manin-Segre rigidity, and the Dolgachev-Iskovskikh classification of finite subgroups of Cr2. The authors also rely on two of their own prior papers (Pinardin 2024, Yasinsky 2024) for lemmas on degree 6 del Pezzo surfaces. There are no free parameters or invented entities; the result is a classification theorem within ZFC plus standard algebraic geometry.

assumptions (7)
  • standard math Finite subgroups of PGL2(k) are classified by Klein's list: Cn, Dn, A4, S4, A5, each with one conjugacy class (Proposition 3.2).
    Invoked throughout Sections 5 and 6 to enumerate possible H in fibre products and orbit lengths on P1.
  • standard math Blichfeldt's classification of finite subgroups of PGL3(k): six primitive types plus intransitive and imprimitive families (Section 3.2).
    Used to state the P2 row of the Main Theorem and to argue that F5 and S5 do not embed into PGL3.
  • domain assumption The equivariant Sarkisov program: every G-birational map between G-Mori fibre spaces decomposes into G-links of types I-IV, with Iskovskikh's complete classification of links in dimension 2 ([Isk96, Theorem 2.6], [DI09, Propositions 7.12, 7.13]).
    This is the load-bearing machinery; non-linearizability claims rely on exhausting all possible links.
  • standard math Manin-Segre rigidity: G-del Pezzo surfaces with K^2 ≤ 3 are G-birationally rigid, and K^2 = 1 is superrigid (Theorem 2.3); G-conic bundles with K^2 ≤ 0 are G-birationally superrigid (Theorem 2.4).
    Reduces the problem to K^2 ≥ 4 del Pezzo surfaces and no-singular-fibre conic bundles.
  • domain assumption Dolgachev-Iskovskikh classification of finite subgroups of Cr2(k) and the correspondence between conjugacy classes and G-birational equivalence classes of G-Mori fibre spaces ([DI09]).
    The paper uses DI09's list of possible groups on the quintic del Pezzo surface and the general correspondence between conjugacy classes and G-varieties.
  • domain assumption For the quintic del Pezzo surface, A5 and S5 are G-birationally superrigid ([Che08, Example 6.3], [Che14, Theorem B.10]); for the sextic del Pezzo surface, the groups C6, S3, D6 and torus extensions are classified by [Yas24] and [Pin24b] (self-citations).
    These are external classification results the proof imports to restrict the possible groups and their rigidity.
  • domain assumption Linearizability of a G-action on F0 is equivalent to the existence of a sequence of G-elementary transformations and type IV links from F0 to F1, followed by the blow-down of the unique (-1)-curve ([DI09, Propositions 7.12, 7.13]).
    Used in Section 6.3 to reduce Theorem 6.14 to orbit arithmetic and the Euclidean algorithm.

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Pith. "Pith review of Linearization problem for finite subgroups of the plane Cremona group." pith.science (2026). https://pith.science/paper/AGULDHTY

@misc{pith2026241212022,
  author       = {Pith},
  title        = {Pith review of: Linearization problem for finite subgroups of the plane Cremona group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AGULDHTY}},
  note         = {Machine review of arXiv:2412.12022}
}
read the original abstract

We give a complete solution of the linearization problem in the plane Cremona group over an algebraically closed field of characteristic zero.

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