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arxiv: 2607.00652 · v1 · pith:2KSAB4O7new · submitted 2026-07-01 · 🧮 math.AG

Automorphism groups of non-normal rigid affine surfaces are finite-dimensional

Ivan Beldiev , Alexander Perepechko This is my paper

Pith reviewed 2026-07-02 06:19 UTC · model grok-4.3

classification 🧮 math.AG
keywords automorphism groupaffine surfacenon-normalrigidfinite-dimensionaladditive group actionalgebraic geometry
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The pith

The automorphism group of a non-normal affine surface is finite-dimensional if and only if it admits no non-trivial additive group action.

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

This paper extends a previous result on normal affine surfaces to the non-normal case. It establishes that the automorphism group is finite-dimensional precisely when the surface has no non-trivial action by the additive group. A reader would care because this provides a criterion for when these groups are finite-dimensional without requiring the surface to be normal. The extension relies on rigidity to ensure the properties transfer.

Core claim

The authors extend the equivalence established for normal affine surfaces: the automorphism group of a non-normal affine surface is finite-dimensional if and only if the surface admits no non-trivial action of the additive group of the base field. This is shown for rigid non-normal affine surfaces.

What carries the argument

The equivalence between finite-dimensionality of the automorphism group and absence of non-trivial Ga-actions on the surface, extended from normal to non-normal cases using rigidity.

If this is right

  • The result classifies non-normal rigid affine surfaces with finite-dimensional automorphism groups.
  • It broadens the applicability of the criterion to surfaces that are not normal.
  • Automorphism groups remain finite-dimensional under the no non-trivial additive action condition even without normality.

Where Pith is reading between the lines

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

  • This suggests the criterion may hold for other classes of surfaces beyond affine ones.
  • Further work could test if rigidity is necessary or if the result generalizes to higher dimensions.

Load-bearing premise

The definitions and properties of rigidity and finite-dimensionality extend to non-normal surfaces without introducing new obstructions that would break the equivalence.

What would settle it

Finding a non-normal affine surface with no non-trivial additive group action whose automorphism group is nevertheless infinite-dimensional would disprove the claim.

read the original abstract

It was recently established by Perepechko and Zaidenberg that the automorphism group of a normal affine surface is finite-dimensional if and only if the surface admits no non-trivial action of the additive group of the base field. We extend this result to non-normal affine surfaces.

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 / 1 minor

Summary. The manuscript extends the theorem of Perepechko and Zaidenberg, which characterizes finite-dimensional automorphism groups of normal affine surfaces by the absence of non-trivial Ga-actions, to the setting of non-normal affine surfaces. The central claim is that the same equivalence holds without the normality assumption.

Significance. If the extension is valid, the result supplies a uniform criterion for finite-dimensionality of Aut(X) that applies to all affine surfaces, normal or not. This would strengthen the link between rigidity (no Ga-action) and the structure of automorphism groups in the broader category of affine surfaces.

minor comments (1)
  1. The abstract states the extension but provides no indication of how the definitions of rigidity and finite-dimensionality are adapted to non-normal surfaces or how the proof avoids new obstructions at singularities; a brief outline in the introduction would clarify the scope of the extension.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for summarizing our extension of the Perepechko-Zaidenberg theorem to non-normal affine surfaces. The central claim is that finite-dimensionality of Aut(X) is equivalent to the absence of non-trivial Ga-actions, and this holds without assuming normality.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The provided abstract cites a prior result by Perepechko and Zaidenberg for the normal case and states an extension to non-normal surfaces, but contains no equations, definitions, or derivations that reduce the claimed extension to the cited result by construction. No self-definitional steps, fitted inputs presented as predictions, or load-bearing self-citations that collapse the central claim are present or quotable. The derivation chain for the extension is not shown to be equivalent to its inputs, making the work self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard definitions and results in algebraic geometry over an algebraically closed field; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math Standard axioms and definitions of algebraic geometry (affine varieties, normality, automorphism groups, Ga-actions) over an algebraically closed field.
    Implicit background for any result in math.AG; invoked when the authors refer to normal vs non-normal surfaces and Ga-actions.

pith-pipeline@v0.9.1-grok · 5557 in / 1087 out tokens · 27024 ms · 2026-07-02T06:19:19.049157+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

12 extracted references · 3 canonical work pages · 1 internal anchor

  1. [1]

    I. V. Arzhantsev and S. A. Gaifullin. The automorphism group of a rigid affine variety.Math. Nachr., 290:662–671, 2017

    2017

  2. [2]

    Boldyrev and S

    I. Boldyrev and S. Gaifullin. Automorphisms of nonnormal toric varieties.Math. Notes, 110:872–886, 2021.doi:10.1134/S0001434621110237

    work page doi:10.1134/s0001434621110237 2021

  3. [3]

    Borovik and S

    V. Borovik and S. Gaifullin. Isolated torus invariants and automorphism groups of rigid vari- eties.J. Algebra, 666:821–839, 2025

    2025

  4. [4]

    Flenner and M

    H. Flenner and M. Zaidenberg. Locally nilpotent derivations on affine surfaces with a∗-action. Osaka J. Math., 42:931–974, 2006

    2006

  5. [5]

    Freudenburg.Algebraic Theory of Locally Nilpotent Derivations, volume 136 ofEncyclopae- dia of Mathematical Sciences

    G. Freudenburg.Algebraic Theory of Locally Nilpotent Derivations, volume 136 ofEncyclopae- dia of Mathematical Sciences. Springer-Verlag, Berlin, 2006

    2006

  6. [6]

    On the geometry of the automorphism groups of affine varieties

    J.-Ph. Furter and H. Kraft. On the geometry of automorphism groups of affine varieties. arXiv:1809.04175, 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  7. [7]

    S. Iitaka. On logarithmic kodaira dimension of algebraic varieties.Compl. Anal. and Algebr. Geom., pages 175–189, 1977

    1977

  8. [8]

    H. Kraft. Automorphism groups of affine varieties and a characterization of affine n-space. Trans. Mosc. Math. Soc., 78:171–186, 2017

    2017

  9. [9]

    H. Kraft. Automorphism groups of varieties – a survey. InMathematisches Forschungsinstitut Oberwolfach, number 20/2025, pages 962–970. 2025. URL:https://publications.mfo.de/ bitstream/handle/mfo/4333/OWR_2025_20.pdf,doi:10.4171/OWR/2025/20

    work page doi:10.4171/owr/2025/20 2025

  10. [10]

    Perepechko and A

    A. Perepechko and A. Regeta. When is the automorphism group of an affine variety nested? Transform. groups, 28:401–412, 2023. AUTOMORPHISM GROUPS OF NON-NORMAL RIGID AFFINE SURF ACES 7

    2023

  11. [11]

    Perepechko and A

    A. Perepechko and A. Regeta. Automorphism groups of affine varieties without non-algebraic elements.Proc. Amer. Math. Soc., 152, 6:2377–2383, 2024

    2024

  12. [12]

    Automorphism groups of rigid affine surfaces: the identity component.Algebraic Geometry, 13:to appear, 2026

    Alexander Perepechko and Mikhail Zaidenberg. Automorphism groups of rigid affine surfaces: the identity component.Algebraic Geometry, 13:to appear, 2026. Email address:ivbeldiev@gmail.com, isbeldiev@hse.ru HSE University, F aculty of Computer Science, Pokrovsky bl vd. 11, Moscow, 109028 Russia Email address:a@perep.ru HSE University, F aculty of Computer ...

    2026