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arxiv: 2604.07519 · v1 · submitted 2026-04-08 · 🧮 math.AG

A one-step counterexample to the normalized Nash blowup conjecture

Alvaro Liendo , Ana Julisa Palomino , Gonzalo Rodr\'iguez This is my paper

Pith reviewed 2026-05-10 17:00 UTC · model grok-4.3

classification 🧮 math.AG
keywords toric varietiesNash blowupcounterexamplesingularitiescharacteristic threenormalizationaffine variety
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The pith

A normal singular five-dimensional toric variety over a field of characteristic three has its normalized Nash blowup already containing an open affine subset isomorphic to the variety itself.

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

The paper constructs an explicit normal singular affine toric variety X of dimension five over an algebraically closed field of characteristic three. It shows that applying the normalized Nash blowup to X produces a variety whose normalization already includes an open affine chart isomorphic to X. This creates an immediate loop after one iteration. Combined with earlier examples, the result gives one-step counterexamples in every dimension at least five and every characteristic. The characteristic-three case is shown to be delicate because the known four-dimensional counterexample required two iterations.

Core claim

We construct an explicit normal singular affine toric variety X of dimension five over an algebraically closed field of characteristic three such that the normalized Nash blowup of X already contains an open affine subset isomorphic to X. Combined with previously known examples, this yields one-step counterexamples in every dimension greater than or equal to five and every characteristic. The characteristic-three case is the most delicate: the previously known counterexample in dimension four requires a two-step iteration of the normalized Nash blowup, and our example demonstrates that in dimension five and higher the minimal number of iterations needed to produce a loop is one.

What carries the argument

The normalized Nash blowup of an affine toric variety, computed from the combinatorial data of its fan and then normalized, which here produces an open chart isomorphic to the original variety.

Load-bearing premise

The explicitly constructed toric variety is normal and singular, and the fan data in characteristic three yields a normalized Nash blowup containing an open affine chart isomorphic to the original variety.

What would settle it

A direct computation of the fan for the normalized Nash blowup showing that no open affine subset is isomorphic to the original variety X.

read the original abstract

We construct an explicit normal singular affine toric variety X of dimension five over an algebraically closed field of characteristic three such that the normalized Nash blowup of X already contains an open affine subset isomorphic to X. Combined with previously known examples, this yields one-step counterexamples in every dimension greater than or equal to five and every characteristic. The characteristic-three case is the most delicate: the previously known counterexample in dimension four requires a two-step iteration of the normalized Nash blowup, and our example demonstrates that in dimension five and higher the minimal number of iterations needed to produce a loop is one.

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 / 0 minor

Summary. The manuscript constructs an explicit normal singular affine toric variety X of dimension five over an algebraically closed field of characteristic three such that the normalized Nash blowup of X contains an open affine subset isomorphic to X. Combined with prior examples, this yields one-step counterexamples to the normalized Nash blowup conjecture in every dimension at least five and every characteristic, with the characteristic-three case shown to require only a single iteration (unlike the two-step example known in dimension four).

Significance. If the explicit construction and its combinatorial verification hold, the result is significant: it completes the picture of one-step counterexamples across all dimensions >=5 and all characteristics. The toric setting permits direct combinatorial checks of normality, singularity, and the Nash transform, which is a methodological strength. The work demonstrates that the minimal number of normalized Nash blowup iterations needed to produce a loop drops to one in dimension five and higher.

major comments (1)
  1. The central claim rests on an explicit fan in dimension 5 whose normalized Nash blowup computation in characteristic 3 produces a chart isomorphic to X. The manuscript must supply the complete list of ray generators (or lattice points) and a transparent, step-by-step outline of the Nash-cone calculation so that the isomorphism and the characteristic-3 arithmetic can be independently verified; without this level of detail the load-bearing verification cannot be confirmed from the text alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the recommendation of minor revision. We appreciate the recognition of the result's significance in completing the picture of one-step counterexamples to the normalized Nash blowup conjecture. We address the single major comment below.

read point-by-point responses

  1. Referee: The central claim rests on an explicit fan in dimension 5 whose normalized Nash blowup computation in characteristic 3 produces a chart isomorphic to X. The manuscript must supply the complete list of ray generators (or lattice points) and a transparent, step-by-step outline of the Nash-cone calculation so that the isomorphism and the characteristic-3 arithmetic can be independently verified; without this level of detail the load-bearing verification cannot be confirmed from the text alone.

    Authors: We agree that a fully explicit and self-contained presentation of the fan and the Nash-cone computation will strengthen the manuscript. In the revised version we will add the complete list of ray generators of the fan defining X together with a transparent, step-by-step outline of the normalized Nash blowup calculation in characteristic 3. This will include the explicit lattice-point data, the cones of the Nash fan, the characteristic-3 arithmetic that produces the fixed chart, and the verification that the resulting open affine subset is isomorphic to X. revision: yes

Circularity Check

0 steps flagged

Explicit combinatorial construction with no reduction to inputs

full rationale

The paper supplies an explicit toric fan in dimension 5 over an algebraically closed field of characteristic 3, verifies that the associated affine variety is normal and singular directly from the fan data, and performs a direct combinatorial computation of the normalized Nash blowup cones to exhibit an open affine chart isomorphic to the original variety. This constitutes a self-contained construction and verification; the central claim does not reduce by definition or by fitted parameters to its own inputs, nor does it rest on a load-bearing self-citation chain. Prior lower-dimensional examples are cited only for context and to combine results, but the dimension-5 case stands independently once the fan and cone computations are accepted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on the standard theory of toric varieties and the definition of the normalized Nash blowup; no free parameters, new entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)
  • standard math Toric varieties are determined by fans in lattices and their properties such as normality and singularity can be read combinatorially.
    The paper uses this established framework to define the variety and compute its Nash blowup.
  • standard math The normalized Nash blowup is well-defined for normal toric varieties over algebraically closed fields of any characteristic.
    This is the operation whose iteration is studied in the conjecture.

pith-pipeline@v0.9.0 · 5396 in / 1520 out tokens · 73900 ms · 2026-05-10T17:00:29.888019+00:00 · methodology

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    Resolving toric varieties with N ash blowups

    Atanas Atanasov, Christopher Lopez, Alexander Perry, Nicholas Proudfoot, and Michael Thaddeus. Resolving toric varieties with N ash blowups. Exp. Math. , 20(3):288--303, 2011

    work page 2011

  2. [2]

    Characteristic-free normalized nash blowup of toric varieties

    Federico Castillo, Daniel Duarte, Maximiliano Leyton-\'Alvarez, and Alvaro Liendo. Characteristic-free normalized nash blowup of toric varieties. arXiv:2501.09811 , 2025

    work page arXiv 2025

  3. [3]

    Non-normalized nash blowup fails to resolve singularities in dimension three

    Federico Castillo, Daniel Duarte, Maximiliano Leyton-\'Alvarez, and Alvaro Liendo. Non-normalized nash blowup fails to resolve singularities in dimension three. arXiv:2511.01772 , 2025

    work page arXiv 2025

  4. [4]

    On a computational approach to the nash blowup problem

    Federico Castillo, Daniel Duarte, Maximiliano Leyton-\'Alvarez, and Alvaro Liendo. On a computational approach to the nash blowup problem. arXiv:2511.17862 , 2025

    work page arXiv 2025

  5. [5]

    Nash blowup fails to resolve singularities in dimensions four and higher

    Federico Castillo, Daniel Duarte, Maximiliano Leyton-\'Alvarez, and Alvaro Liendo. Nash blowup fails to resolve singularities in dimensions four and higher. Annals of Mathematics , 203(2):677--694, March 2026

    work page 2026

  6. [6]

    Cox, John B

    David A. Cox, John B. Little, and Henry K. Schenck. Toric varieties , volume 124. American Mathematical Soc., 2011

    work page 2011

  7. [7]

    Dalbelo, Daniel Duarte, and Maria Aparecida Soares Ruas

    Tha\'is M. Dalbelo, Daniel Duarte, and Maria Aparecida Soares Ruas. Nash blowups of 2-generic determinantal varieties in positive characteristic. Pure Appl. Math. Q. , 21(4):1557--1575, 2025

    work page 2025

  8. [8]

    Nash blowups of toric varieties in prime characteristic

    Daniel Duarte, Jack Jeffries, and Luis N\'u\ nez Betancourt. Nash blowups of toric varieties in prime characteristic. Collect. Math. , 75(3):629--637, 2024

    work page 2024

  9. [9]

    Nash blowups in prime characteristic

    Daniel Duarte and Luis N\'u\ nez Betancourt. Nash blowups in prime characteristic. Rev. Mat. Iberoam. , 38(1):257--267, 2022

    work page 2022

  10. [10]

    Nash modification on toric curves

    Daniel Duarte and Daniel Green Tripp. Nash modification on toric curves. In Singularities, algebraic geometry, commutative algebra, and related topics , pages 191--202. Springer, Cham, 2018

    work page 2018

  11. [11]

    Nash modification on toric surfaces

    Daniel Duarte. Nash modification on toric surfaces. Rev. R. Acad. Cienc. Exactas F\' s. Nat. Ser. A Mat. RACSAM , 108(1):153--171, 2014

    work page 2014

  12. [12]

    Introduction to toric varieties

    William Fulton. Introduction to toric varieties . Number 131 in Annals of mathematics studies. Princeton university press, 1993

    work page 1993

  13. [13]

    Dima Grigoriev and Pierre D. Milman. Nash resolution for binomial varieties as E uclidean division. A priori termination bound, polynomial complexity in essential dimension 2. Adv. Math. , 231(6):3389--3428, 2012

    work page 2012

  14. [14]

    Gonz\'alez P\'erez and Bernard Teissier

    Pedro D. Gonz\'alez P\'erez and Bernard Teissier. Toric geometry and the S emple- N ash modification. Rev. R. Acad. Cienc. Exactas F\'is. Nat. Ser. A Mat. RACSAM , 108(1):1--48, 2014

    work page 2014

  15. [15]

    \' E ventails en dimension 2 et transform\' e de N ash

    Gerardo Gonzalez Sprinberg. \' E ventails en dimension 2 et transform\' e de N ash. Publications du Centre de Math\' e matiques de l'E.N.S. , pages 1--68, 1977

    work page 1977

  16. [16]

    R\' e solution de N ash des points doubles rationnels

    Gerardo Gonzalez-Sprinberg. R\' e solution de N ash des points doubles rationnels. Ann. Inst. Fourier (Grenoble) , 32(2):x, 111--178, 1982

    work page 1982

  17. [17]

    On N ash blow-up of orbifolds

    Gerard Gonzalez-Sprinberg. On N ash blow-up of orbifolds. In Singularities--- N iigata-- T oyama 2007 , volume 56 of Adv. Stud. Pure Math. , pages 133--149. Math. Soc. Japan, Tokyo, 2009

    work page 2007

  18. [18]

    On N ash blowing-up

    Heisuke Hironaka. On N ash blowing-up. In Arithmetic and geometry, V ol. II , volume 36 of Progr. Math. , pages 103--111. Birkh\"auser, Boston, MA, 1983

    work page 1983

  19. [19]

    A. Nobile. Some properties of the N ash blowing-up. Pacific J. Math. , 60(1):297--305, 1975

    work page 1975

  20. [20]

    Convex bodies and algebraic geometry: an introduction to the theory of toric varieties

    Tadao Oda. Convex bodies and algebraic geometry: an introduction to the theory of toric varieties . Springer, 1983

    work page 1983

  21. [21]

    Desingularization properties of the N ash blowing-up process, 1977

    Vaho Rebassoo. Desingularization properties of the N ash blowing-up process, 1977. Ph. D. dissertation, University of Washington

    work page 1977

  22. [22]

    S ageMath, the S age M athematics S oftware S ystem ( V ersion 10.4) , 2024

    Sage Developers . S ageMath, the S age M athematics S oftware S ystem ( V ersion 10.4) , 2024. https://www.sagemath.org

    work page 2024

  23. [23]

    J. G. Semple. Some investigations in the geometry of curve and surface elements. Proc. London Math. Soc. (3) , 4:24--49, 1954

    work page 1954

  24. [24]

    Sandwiched singularities and desingularization of surfaces by normalized N ash transformations

    Mark Spivakovsky. Sandwiched singularities and desingularization of surfaces by normalized N ash transformations. Ann. of Math. (2) , 131(3):411--491, 1990

    work page 1990

  25. [25]

    Resolution of singularities: an introduction

    Mark Spivakovsky. Resolution of singularities: an introduction. In Handbook of geometry and topology of singularities. I , pages 183--242. Springer, Cham, [2020] 2020

    work page 2020