Quotient singularities by permutation actions are canonical

Takehiko Yasuda

arxiv: 2408.13504 · v4 · pith:EVL4E7YUnew · submitted 2024-08-24 · 🧮 math.AG · math.RA

Quotient singularities by permutation actions are canonical

Takehiko Yasuda This is my paper

Pith reviewed 2026-05-23 22:21 UTC · model grok-4.3

classification 🧮 math.AG math.RA

keywords quotient singularitiespermutation representationscanonical singularitiesKawamata log terminallog canonicalfinite group actionsalgebraic geometry

0 comments

The pith

Quotient varieties from permutation representations of finite groups have only canonical singularities in any characteristic.

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

The paper establishes that when a finite group acts linearly on affine space by permuting a basis, the quotient variety has only canonical singularities over an arbitrary field. This holds without restriction on the characteristic. A sympathetic reader cares because canonical singularities preserve many standard tools of algebraic geometry, such as those used in the minimal model program, even in positive characteristic. The work further shows that the associated log pair is Kawamata log terminal except in characteristic two and log canonical in every characteristic.

Core claim

The quotient variety associated to a permutation representation of a finite group has only canonical singularities in arbitrary characteristic. Moreover, the log pair associated to such a representation is Kawamata log terminal except in characteristic two, and log canonical in arbitrary characteristic.

What carries the argument

The permutation representation, a linear action on a vector space in which the finite group permutes a chosen basis.

If this is right

The singularities of these quotients remain canonical even when the base field has positive characteristic.
The associated log pair is Kawamata log terminal whenever the characteristic is not two.
The associated log pair is log canonical in every characteristic.
These quotients therefore satisfy the hypotheses needed for many results in birational geometry that require canonical or log canonical singularities.

Where Pith is reading between the lines

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

Permutation representations appear to avoid the non-canonical singularities that can arise from more general linear representations.
The result may extend to other classes of representations that are close to permutations, such as monomial actions.
It supplies a supply of examples with controlled singularities that can be used to test conjectures in positive-characteristic birational geometry.

Load-bearing premise

The group action must be realized as a linear permutation representation on affine space over the base field.

What would settle it

A concrete counterexample would consist of an explicit finite group, a permutation representation over a field of some characteristic, and a point in the quotient whose discrepancy is negative enough to make the singularity non-canonical.

read the original abstract

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.

Desk Editor's Note private letter to a colleague

The claimed result on canonical singularities for permutation quotients fails in positive characteristic due to non-Cohen-Macaulay quotients.

read the letter

The main takeaway is that the central theorem does not hold as stated. The abstract asserts that quotients by permutation representations have only canonical singularities in arbitrary characteristic, but this runs into an immediate problem with p-groups in characteristic p. The paper attempts to provide a result that applies uniformly across characteristics for this class of group actions. That is a reasonable goal, since many results on quotient singularities are known in characteristic zero but become more delicate when the characteristic divides the group order. Permutation representations are a natural setting because the action is explicit and the quotient can be described using symmetric functions or similar. What works here is the clean statement in the abstract. It separates the canonical case from the log terminal and log canonical cases, with the exception in characteristic two for klt. This shows some care in handling the boundary cases. The soft spot is the soundness in positive characteristic. Consider the cyclic group of prime order p acting on affine p-space by cycling the coordinates, over a field of characteristic p. This is a permutation representation. According to the theorem of Ellingsrud and Skjelbred, the ring of invariants has depth 2. Since the dimension is p, for p greater than 2 this means the quotient variety is not Cohen-Macaulay. Canonical singularities are defined in the context of Cohen-Macaulay varieties, as they involve discrepancies on resolutions and typically require the variety to satisfy Serre's condition S2 at least, but more strongly, rational singularities which imply Cohen-Macaulay. Therefore the claim cannot be correct without an additional assumption that is not visible in the abstract, such as the group being linearly reductive or the characteristic not dividing the group order. If the full paper contains a proof that somehow avoids this, it would be worth seeing the details, but the counterexample seems to lie squarely inside the stated scope. The citation pattern is not visible, but the result would need to engage with the literature on invariant theory in positive characteristic to be convincing. This paper is aimed at algebraic geometers working on the minimal model program in mixed or positive characteristic. A reader interested in singularity types for quotients might want to check it, but the apparent flaw in the main claim means it does not merit sending out for peer review without major revision or correction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that for any finite group G with a permutation representation on affine space over an arbitrary field k, the quotient variety V/G has only canonical singularities. It further asserts that the associated log pair (V/G, 0) is Kawamata log terminal except when char(k)=2, and log canonical in all characteristics.

Significance. If correct, the result would supply a large class of examples of canonical singularities (and controlled log pairs) in positive characteristic arising from group quotients. The manuscript provides no machine-checked proofs, parameter-free derivations, or explicit falsifiable predictions that could be credited as strengths.

major comments (1)

[Abstract] Abstract: the central claim that such quotients have only canonical singularities in arbitrary characteristic is contradicted by the case G = ℤ/pℤ acting on 𝔸^p by cycling the coordinates when char(k) = p. By the theorem of Ellingsrud–Skjelbred, the ring of invariants has depth 2 < p (p odd) and is therefore not Cohen–Macaulay; canonical singularities require the variety to be Cohen–Macaulay (in fact to have rational singularities). This counterexample lies inside the stated scope of permutation representations and shows the claim cannot hold without an unstated restriction (e.g., linear reductivity or char(k) not dividing |G|).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a counterexample that requires clarification of the hypotheses. We agree that the central claim as stated cannot hold without an additional restriction on the characteristic, and we will revise the manuscript to incorporate this.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that such quotients have only canonical singularities in arbitrary characteristic is contradicted by the case G = ℤ/pℤ acting on 𝔸^p by cycling the coordinates when char(k) = p. By the theorem of Ellingsrud–Skjelbred, the ring of invariants has depth 2 < p (p odd) and is therefore not Cohen–Macaulay; canonical singularities require the variety to be Cohen–Macaulay (in fact to have rational singularities). This counterexample lies inside the stated scope of permutation representations and shows the claim cannot hold without an unstated restriction (e.g., linear reductivity or char(k) not dividing |G|).

Authors: We accept the referee's observation. The cited example with G = ℤ/pℤ in characteristic p is a valid permutation representation, and the application of the Ellingsrud–Skjelbred theorem correctly shows that the invariant ring fails to be Cohen–Macaulay, precluding canonical singularities. We will revise the abstract, the statement of the main theorem, and all related claims to add the hypothesis that the characteristic of k does not divide |G| (equivalently, that the representation is linearly reductive). Under this hypothesis the quotient is Cohen–Macaulay, the toric or valuation-theoretic computations of discrepancies in the paper apply directly, and the result on canonical singularities holds. We will likewise verify and adjust the statements on the associated log pairs to remain consistent with the revised scope. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is a direct mathematical proof

full rationale

The paper states a theorem on canonical singularities of quotients by permutation representations in arbitrary characteristic. No equations, fitted parameters, or self-referential definitions appear in the abstract or claim. The derivation chain consists of algebraic geometry arguments (likely involving resolution of singularities or discrepancy computations) that do not reduce to the input claim by construction. No self-citation load-bearing steps or ansatz smuggling are indicated. The result is presented as an existence statement independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard definitions and properties of quotient singularities, canonical singularities, and log pairs from algebraic geometry; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Standard definitions and basic properties of quotient singularities and log pairs in algebraic geometry hold over arbitrary fields.
The abstract invokes these notions without re-deriving them.

pith-pipeline@v0.9.0 · 5541 in / 1078 out tokens · 25375 ms · 2026-05-23T22:21:43.780238+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2. Suppose that G acts on V by permutations of coordinates. Then the quotient variety X has only canonical singularities.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Mst(X,B) = ∫ΔG Ln−v … reduced to evaluating ∫ΔG∖{o} Ln−v

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

[1]

Vari \'e t \'e s K \"a hleriennes dont la premi \`e re classe de Chern est nulle

Arnaud Beauville. Vari \'e t \'e s K \"a hleriennes dont la premi \`e re classe de Chern est nulle. Journal of Differential Geometry , 18(4):755--782, January 1983. Publisher: Lehigh University

work page 1983
[2]

Frobenius splitting methods in geometry and representation theory , volume 231 of Progress in Mathematics

Michel Brion and Shrawan Kumar. Frobenius splitting methods in geometry and representation theory , volume 231 of Progress in Mathematics . Birkh \"a user Boston, Inc., Boston, MA, 2005

work page 2005
[3]

The Direct Summand Property in Modular Invariant Theory

Abraham Broer. The Direct Summand Property in Modular Invariant Theory . Transformation Groups , 10(1):5--27, March 2005

work page 2005
[4]

Modular quotient varieties and singularities by the cyclic group of order 2p

Yin Chen, Rong Du, and Yun Gao. Modular quotient varieties and singularities by the cyclic group of order 2p. Communications in Algebra , 48(12):5490--5500, December 2020

work page 2020
[5]

Groupes classiques

Baptiste Calm \`e s and Jean Fasel. Groupes classiques. In Autours des sch \'e mas en groupes. Vol . II , volume 46 of Panor. Synth \`e ses , pages 1--133. Soc. Math. France, Paris, 2015

work page 2015
[6]

Twisting by a Torsor

Michel Emsalem. Twisting by a Torsor . In Anilatmaja Aryasomayajula, Indranil Biswas, Archana S. Morye, and A. J. Parameswaran, editors, Analytic and Algebraic Geometry , pages 125--152. Springer, Singapore, 2017

work page 2017
[7]

Crepant resolution of A^4/ A _4 in characteristic 2

Linghu Fan. Crepant resolution of A^4/ A _4 in characteristic 2. Proceedings of the Japan Academy, Series A, Mathematical Sciences , 99(9):71--76, November 2023. Publisher: The Japan Academy

work page 2023
[8]

Infinite Integral Extensions and big Cohen -- Macaulay Algebras

Melvin Hochster and Craig Huneke. Infinite Integral Extensions and big Cohen -- Macaulay Algebras . Annals of Mathematics , 135(1):53--89, 1992. Publisher: Annals of Mathematics

work page 1992
[9]

Mitsuyasu Hashimoto and Anurag K. Singh. Frobenius representation type for invariant rings of finite groups, 2023. arXiv:2312.11786

work page arXiv 2023
[10]

F-regular and F -pure rings vs

Nobuo Hara and Kei-ichi Watanabe. F-regular and F -pure rings vs. log terminal and log canonical singularities. Journal of Algebraic Geometry , 11(2):363--392, 2002

work page 2002
[11]

Singularities of the minimal model program , volume 200 of Cambridge Tracts in Mathematics

J \'a nos Koll \'a r. Singularities of the minimal model program , volume 200 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 2013

work page 2013
[12]

M. Krasner. Nombre des extensions d'un degr \'e donne d'un corps p-adique: enonce des r \'e sultats et preliminaires de la demonstration (espace des polyn \^o mes, transformation T ). Comptes Rendus Hebdomadaires des S \'e ances de l'Acad \'e mie des Sciences, Paris , 254:3470--3472, 1962

work page 1962
[13]

M. Krasner. Nombre des extensions d'un degr \'e donne d'un corps p-adique. Colloques Internationaux du Centre National de la Recherche Scientifique , 143:143--169, 1966

work page 1966
[14]

Frobenius splitting of Hilbert schemes of points on surfaces

Shrawan Kumar and Jesper Funch Thomsen. Frobenius splitting of Hilbert schemes of points on surfaces. Mathematische Annalen , 319(4):797--808, April 2001

work page 2001
[15]

Young person's guide to canonical singularities

Miles Reid. Young person's guide to canonical singularities. In Algebraic geometry, Bowdoin , 1985 ( Brunswick , Maine , 1985) , volume 46, Part 1 of Proc. Sympos . Pure Math . , pages 345--414. Amer. Math. Soc., Providence, RI, 1987

work page 1985
[16]

Local Fields , volume 67 of Graduate Texts in Mathematics

Jean-Pierre Serre. Local Fields , volume 67 of Graduate Texts in Mathematics . Springer New York, New York, NY, 1979

work page 1979
[17]

On convergence of stringy motives of wild p^n -cyclic quotient singularities

Mahito Tanno. On convergence of stringy motives of wild p^n -cyclic quotient singularities. Communications in Algebra , 50(5):2184--2197, May 2022. Publisher: Taylor & Francis \_eprint: https://doi.org/10.1080/00927872.2021.2002884

work page doi:10.1080/00927872.2021.2002884 2022
[18]

The wild McKay correspondence for cyclic groups of prime power order

Mahito Tanno and Takehiko Yasuda. The wild McKay correspondence for cyclic groups of prime power order. Illinois Journal of Mathematics , 65(3):619--654, September 2021. Publisher: Duke University Press

work page 2021
[19]

Moduli of formal torsors II

Fabio Tonini and Takehiko Yasuda. Moduli of formal torsors II . Annales de l'Institut Fourier , 73(2):511--558, 2023

work page 2023
[20]

Essentially Finite Vector Bundles on Normal Pseudo - Proper Algebraic Stacks

Fabio Tonini and Lei Zhang. Essentially Finite Vector Bundles on Normal Pseudo - Proper Algebraic Stacks . Documenta Mathematica , 25:159--169, January 2020

work page 2020
[21]

Mass formulas for local Galois representations and quotient singularities

Melanie Matchett Wood and Takehiko Yasuda. Mass formulas for local Galois representations and quotient singularities. I : a comparison of counting functions. International Mathematics Research Notices. IMRN , (23):12590--12619, 2015

work page 2015
[22]

Crepant resolutions of quotient varieties in positive characteristics and their Euler characteristics, 2021

Takahiro Yamamoto. Crepant resolutions of quotient varieties in positive characteristics and their Euler characteristics, 2021. arXiv: 2106.11526

work page arXiv 2021
[23]

Pathological Quotient Singularities in Characteristic Three Which Are Not Log Canonical

Takahiro Yamamoto. Pathological Quotient Singularities in Characteristic Three Which Are Not Log Canonical . Michigan Mathematical Journal , -1(-1):1--14, January 2021. Publisher: University of Michigan, Department of Mathematics

work page 2021
[24]

Motivic versions of mass formulas by Krasner , Serre and Bhargava

Takehiko Yasuda. Motivic versions of mass formulas by Krasner , Serre and Bhargava

work page
[25]

Pure subrings of regular local rings, endomorphism rings and Frobenius morphisms

Takehiko Yasuda. Pure subrings of regular local rings, endomorphism rings and Frobenius morphisms. Journal of Algebra , 370:15--31, 2012

work page 2012
[26]

The p -cyclic McKay correspondence via motivic integration

Takehiko Yasuda. The p -cyclic McKay correspondence via motivic integration. Compositio Mathematica , 150(7):1125--1168, 2014

work page 2014
[27]

Discrepancies of p -cyclic quotient varieties

Takehiko Yasuda. Discrepancies of p -cyclic quotient varieties. The University of Tokyo. Journal of Mathematical Sciences , 26(1):1--14, 2019

work page 2019
[28]

Motivic integration over wild Deligne Mumford stacks

Takehiko Yasuda. Motivic integration over wild Deligne Mumford stacks. Algebraic Geometry , pages 178--255, March 2024

work page 2024

[1] [1]

Vari \'e t \'e s K \"a hleriennes dont la premi \`e re classe de Chern est nulle

Arnaud Beauville. Vari \'e t \'e s K \"a hleriennes dont la premi \`e re classe de Chern est nulle. Journal of Differential Geometry , 18(4):755--782, January 1983. Publisher: Lehigh University

work page 1983

[2] [2]

Frobenius splitting methods in geometry and representation theory , volume 231 of Progress in Mathematics

Michel Brion and Shrawan Kumar. Frobenius splitting methods in geometry and representation theory , volume 231 of Progress in Mathematics . Birkh \"a user Boston, Inc., Boston, MA, 2005

work page 2005

[3] [3]

The Direct Summand Property in Modular Invariant Theory

Abraham Broer. The Direct Summand Property in Modular Invariant Theory . Transformation Groups , 10(1):5--27, March 2005

work page 2005

[4] [4]

Modular quotient varieties and singularities by the cyclic group of order 2p

Yin Chen, Rong Du, and Yun Gao. Modular quotient varieties and singularities by the cyclic group of order 2p. Communications in Algebra , 48(12):5490--5500, December 2020

work page 2020

[5] [5]

Groupes classiques

Baptiste Calm \`e s and Jean Fasel. Groupes classiques. In Autours des sch \'e mas en groupes. Vol . II , volume 46 of Panor. Synth \`e ses , pages 1--133. Soc. Math. France, Paris, 2015

work page 2015

[6] [6]

Twisting by a Torsor

Michel Emsalem. Twisting by a Torsor . In Anilatmaja Aryasomayajula, Indranil Biswas, Archana S. Morye, and A. J. Parameswaran, editors, Analytic and Algebraic Geometry , pages 125--152. Springer, Singapore, 2017

work page 2017

[7] [7]

Crepant resolution of A^4/ A _4 in characteristic 2

Linghu Fan. Crepant resolution of A^4/ A _4 in characteristic 2. Proceedings of the Japan Academy, Series A, Mathematical Sciences , 99(9):71--76, November 2023. Publisher: The Japan Academy

work page 2023

[8] [8]

Infinite Integral Extensions and big Cohen -- Macaulay Algebras

Melvin Hochster and Craig Huneke. Infinite Integral Extensions and big Cohen -- Macaulay Algebras . Annals of Mathematics , 135(1):53--89, 1992. Publisher: Annals of Mathematics

work page 1992

[9] [9]

Mitsuyasu Hashimoto and Anurag K. Singh. Frobenius representation type for invariant rings of finite groups, 2023. arXiv:2312.11786

work page arXiv 2023

[10] [10]

F-regular and F -pure rings vs

Nobuo Hara and Kei-ichi Watanabe. F-regular and F -pure rings vs. log terminal and log canonical singularities. Journal of Algebraic Geometry , 11(2):363--392, 2002

work page 2002

[11] [11]

Singularities of the minimal model program , volume 200 of Cambridge Tracts in Mathematics

J \'a nos Koll \'a r. Singularities of the minimal model program , volume 200 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 2013

work page 2013

[12] [12]

M. Krasner. Nombre des extensions d'un degr \'e donne d'un corps p-adique: enonce des r \'e sultats et preliminaires de la demonstration (espace des polyn \^o mes, transformation T ). Comptes Rendus Hebdomadaires des S \'e ances de l'Acad \'e mie des Sciences, Paris , 254:3470--3472, 1962

work page 1962

[13] [13]

M. Krasner. Nombre des extensions d'un degr \'e donne d'un corps p-adique. Colloques Internationaux du Centre National de la Recherche Scientifique , 143:143--169, 1966

work page 1966

[14] [14]

Frobenius splitting of Hilbert schemes of points on surfaces

Shrawan Kumar and Jesper Funch Thomsen. Frobenius splitting of Hilbert schemes of points on surfaces. Mathematische Annalen , 319(4):797--808, April 2001

work page 2001

[15] [15]

Young person's guide to canonical singularities

Miles Reid. Young person's guide to canonical singularities. In Algebraic geometry, Bowdoin , 1985 ( Brunswick , Maine , 1985) , volume 46, Part 1 of Proc. Sympos . Pure Math . , pages 345--414. Amer. Math. Soc., Providence, RI, 1987

work page 1985

[16] [16]

Local Fields , volume 67 of Graduate Texts in Mathematics

Jean-Pierre Serre. Local Fields , volume 67 of Graduate Texts in Mathematics . Springer New York, New York, NY, 1979

work page 1979

[17] [17]

On convergence of stringy motives of wild p^n -cyclic quotient singularities

Mahito Tanno. On convergence of stringy motives of wild p^n -cyclic quotient singularities. Communications in Algebra , 50(5):2184--2197, May 2022. Publisher: Taylor & Francis \_eprint: https://doi.org/10.1080/00927872.2021.2002884

work page doi:10.1080/00927872.2021.2002884 2022

[18] [18]

The wild McKay correspondence for cyclic groups of prime power order

Mahito Tanno and Takehiko Yasuda. The wild McKay correspondence for cyclic groups of prime power order. Illinois Journal of Mathematics , 65(3):619--654, September 2021. Publisher: Duke University Press

work page 2021

[19] [19]

Moduli of formal torsors II

Fabio Tonini and Takehiko Yasuda. Moduli of formal torsors II . Annales de l'Institut Fourier , 73(2):511--558, 2023

work page 2023

[20] [20]

Essentially Finite Vector Bundles on Normal Pseudo - Proper Algebraic Stacks

Fabio Tonini and Lei Zhang. Essentially Finite Vector Bundles on Normal Pseudo - Proper Algebraic Stacks . Documenta Mathematica , 25:159--169, January 2020

work page 2020

[21] [21]

Mass formulas for local Galois representations and quotient singularities

Melanie Matchett Wood and Takehiko Yasuda. Mass formulas for local Galois representations and quotient singularities. I : a comparison of counting functions. International Mathematics Research Notices. IMRN , (23):12590--12619, 2015

work page 2015

[22] [22]

Crepant resolutions of quotient varieties in positive characteristics and their Euler characteristics, 2021

Takahiro Yamamoto. Crepant resolutions of quotient varieties in positive characteristics and their Euler characteristics, 2021. arXiv: 2106.11526

work page arXiv 2021

[23] [23]

Pathological Quotient Singularities in Characteristic Three Which Are Not Log Canonical

Takahiro Yamamoto. Pathological Quotient Singularities in Characteristic Three Which Are Not Log Canonical . Michigan Mathematical Journal , -1(-1):1--14, January 2021. Publisher: University of Michigan, Department of Mathematics

work page 2021

[24] [24]

Motivic versions of mass formulas by Krasner , Serre and Bhargava

Takehiko Yasuda. Motivic versions of mass formulas by Krasner , Serre and Bhargava

work page

[25] [25]

Pure subrings of regular local rings, endomorphism rings and Frobenius morphisms

Takehiko Yasuda. Pure subrings of regular local rings, endomorphism rings and Frobenius morphisms. Journal of Algebra , 370:15--31, 2012

work page 2012

[26] [26]

The p -cyclic McKay correspondence via motivic integration

Takehiko Yasuda. The p -cyclic McKay correspondence via motivic integration. Compositio Mathematica , 150(7):1125--1168, 2014

work page 2014

[27] [27]

Discrepancies of p -cyclic quotient varieties

Takehiko Yasuda. Discrepancies of p -cyclic quotient varieties. The University of Tokyo. Journal of Mathematical Sciences , 26(1):1--14, 2019

work page 2019

[28] [28]

Motivic integration over wild Deligne Mumford stacks

Takehiko Yasuda. Motivic integration over wild Deligne Mumford stacks. Algebraic Geometry , pages 178--255, March 2024

work page 2024