Pith. sign in

REVIEW 4 major objections 4 minor 23 references

Infinitely generated symbolic Rees rings of positive characteristic

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For an explicit non-prime ideal in three variables, the extended symbolic Rees ring is Noetherian exactly over fields of characteristic 2 or 3, and infinitely generated over fields of every prime characteristic p ≥ 5.

desk verdict Serious paper with a striking characteristic dichotomy for a non-prime symbolic Rees ring; the char 2/3 half is clean, but the p≥5 half rests on heavy unverified binomial arithmetic that needs referee-level checking. read the letter →

arxiv 2508.04127 v1 pith:RRFNMVTH submitted 2025-08-06 math.AC math.AG

Kazuhiko Kurano This is my paper · ORCID
classification math.ACmath.AG MSC 13A3014M2514C2013A35
keywords symbolicReesringsCoxtoricvarietiespositivecharacteristicMoridreamspacesfinitegenerationspacemonomialcurvesblow-ups
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

The paper studies finite generation of Cox rings of blow-ups of toric surfaces associated to a triangle, in positive characteristic. Its headline example is the ideal I of K[x,y,z] generated by the 2-minors of {{$x^{7}$,$y^{2}$,z},{$y^{11}$,z,$x^{10}$}}; this ideal is not prime, yet its extended symbolic Rees ring R'_s(I) is identified with the Cox ring of the blow-up Y. Applying criteria the paper proves for general triangles (Theorem 1.3), it shows R'_s(I) is Noetherian if and only if the characteristic of K is 2 or 3. This is a complete dichotomy: in every characteristic p ≥ 5 the ring is infinitely generated even though the analogous characteristic-zero ring is finitely generated. The proof works by showing that certain cohomology spaces $H^{0}$(O_Y(-σ j p^r C)|_{σ p^r C}) vanish, which forces the Cox ring to be non-Noetherian.

What carries the argument

The load-bearing identification is Cox(Y) ≅ R'_s(I), the extended symbolic Rees ring of the kernel I of K[x,y,z] → K[Cl(X)]; in the example I is the ideal of 2-minors of the displayed 2×3 matrix. The proof mechanism is the decomposition of ℓC into two affine charts Spec A_ℓ and Spec ψ(B_ℓ), with overlap Spec F_ℓ; the line bundle O(D)|_{ℓC} is encoded by a transition function ξ=(1-x)^u(1-x+vx)^{-u_2}. Theorem 1.3 converts finite generation into the vanishing of $H^{0}$(O_Y(-σ j p^r C)|_{σ p^r C}), and the paper decides this vanishing by a counting argument: certain monomials x_{α,n} span $H^{1}$, and the set D_{p^r,j} of those collapsing modulo A+B must have size at least p^r; Frobenius (p-th power)

What would settle it

For p=13, recompute the coefficient of x_{10e+5,12e+6} obtained by combining (4.19), (4.20), and (4.21) with e=kp+f and f=(p−3)/10; if it is 0 modulo 13, the congruence (4.13) is false and the claimed vanishing $H^{0}$=0 for all r,j collapses. More generally, a direct symbolic check that every displayed coefficient in cases [III-2]–[III-5] is nonzero modulo every relevant p would certify the counting argument; one zero coefficient would locate a p for which the dichotomy could fail.

Watch

Extended reading notes

Core claim

On the author's own terms, the discovery is a set of criteria (Theorem 1.3) that decide, for a toric blow-up Y with a curve C satisfying $C^{2}$=0 and C·E=1, whether Cox(Y) is finitely generated in characteristic p. The criteria are expressed as vanishing of $H^{0}$(O_Y(-σ p^r C)|_{σ p^r C}) or related line-bundle trivializations, where σ is the width of the triangle. For the triangle (4.1) with g=13/6, these criteria give the ideal I = I_2({{$x^{7}$,$y^{2}$,z},{$y^{11}$,z,$x^{10}$}}) with (a,b,c)=(1,1,6); the Cox ring Cox(Y) is R'_s(I), and I is not prime. The paper proves R'_s(I) is Noetherian for p=2 and p=3 by explicitly factoring the transition function $ξ^{2}$ or $ξ^{3}$ in the rings A and ψ(B), and non-Noetherian fo

Load-bearing premise

The non-Noetherian half for every p≥5 rests on a chain of explicit modular congruences showing that certain monomials lie in D_{p^r,j}, and especially on the claim that a displayed binomial coefficient is nonzero modulo p; the printed argument for the p≡3 (mod 10) case does not fully certify that coefficient, and if any link in the chain fails the bound #D_{p^r,j}≥p^r can fail.

Editorial extensions

If this is right

  • The ordinary symbolic Rees ring R_s(I) has the same dichotomy, since R'_s(I) is Noetherian if and only if R_s(I) is.
  • Over any field of characteristic p≥5, the blow-up Y is not a Mori dream space, while over characteristic 2 or 3 it is.
  • For this example, finite generation depends only on the characteristic of the field, not on which particular field of that characteristic is used.
  • The same ideal is finitely generated in characteristic 0, so positive characteristic here genuinely changes the answer from yes to no.
  • For any triangle with W<1, Theorem 1.3 gives finite generation in every positive characteristic; the infinitely generated behavior in this paper is specific to the W=1 boundary case.

Reading between the lines

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

  • The arithmetic of the proof suggests a testable pattern: the primes that divide the binomial coefficients in the p-th power expansion of the transition function ξ may be exactly the primes for which the Cox ring fails to be Noetherian; for this 2×3 matrix that set is {2,3}, but scaling the exponents in the matrix could produce other finite sets of bad primes.
  • A natural next case is to run the same D_{p^r,j} counting on other W=1 triangles; the method should yield, for each triangle, a finite list of characteristics where H^0 can fail to vanish, so the dichotomy 'finite versus infinite generation depends only on membership in a finite set of primes' may be the general phenomenon.
  • If the pattern holds, then positive characteristic is not merely a technical nuisance here: for this example the characteristic-zero ring is finitely generated, so the infinite generation in characteristic p≥5 is a genuinely new phenomenon tied to Frobenius, not inherited from a characteristic-zero obstruction.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies finite generation of Cox rings of blow-ups of toric surfaces associated to a rational triangle, in the case where there is a curve C with C^2 ≤ 0 and C.E = 1. It states criteria for Noetherianity of the Cox ring (Theorems 1.1–1.3), identifies Cox(Y) with the extended symbolic Rees ring R'_s(I) of a generally non-prime ideal I, and then focuses on the triangle with g = 13/6. There the ideal I is generated by the 2-minors of {{x^7, y^2, z}, {y^11, z, x^10}}, and the paper claims that R'_s(I) is Noetherian exactly when the characteristic is 2 or 3. Section 4 gives explicit verifications in characteristics 2 and 3, and a case-by-case arithmetic proof for all p ≥ 5 that R'_s(I) is infinitely generated.

Significance. If the main claim is correct, this is the first complete dichotomy of this kind for a non-prime ideal in the toric setting: an explicit extended symbolic Rees ring that is Noetherian in exactly two positive characteristics and infinitely generated in every characteristic p ≥ 5. The geometric criteria in Theorem 1.3 are natural extensions of earlier work by Cutkosky and by Inagawa–Kurano, and the example is concrete and falsifiable. The paper does not rely on machine-checked computation, but the characteristic 2 and 3 parts are transparent and can be checked by hand. The p ≥ 5 part, however, contains many asserted binomial reductions that are not fully documented; the correctness of these reductions is load-bearing for the dichotomy.

major comments (4)
  1. [Section 3, proof of Theorem 1.1] The equivalence of (A0)–(A3) is asserted to follow 'in the same way' as in Inagawa–Kurano [15], but [15] treats the case where I is prime. In the paper's headline example, d = 24 and I is explicitly not prime. Since Theorem 1.1 is used both for the Noetherian direction in characteristics 2 and 3 and for the implication (C1) ⇒ (C0) in Theorem 1.3, the missing justification is load-bearing. Please either prove the equivalence directly for non-prime I or give a precise statement in [15]/[16] that covers this generality.
  2. [Section 4, [III-3], proof of (4.23)] The derivation of (4.23) depends on the six bracket congruences (4.26)–(4.31) and then on a stated linear combination that is asserted to give a vector whose only nonzero component is the coefficient of x_{10e+5,12e+6}. The intermediate algebra is not shown, and the final polynomial expression in f is not derived. Since a single incorrect binomial or combination coefficient would invalidate (4.23), and hence (4.6), the manuscript should either include the full reduction or provide a reproducible machine verification of these congruences.
  3. [Section 4, [III-5], proof of (4.50)–(4.51)] The analogous situation occurs in the case p ≡ 7 mod 10. The nine-term combination of (4.54)–(4.62) yielding [0,...,0,q1,q2], and the subsequent remainder computations in (4.64)–(4.65), are asserted without derivation. The exceptional prime p = 44777 is treated in a footnote, but the claimed unit status of the relevant denominators is not verified. These computations are essential to the counting bound #D_{p^r,j} ≥ p^r; please supply the missing details or a verifiable computation.
  4. [Section 4, counting and equations (4.12), (4.22), (4.66), (4.67)] The cardinality arguments count pr−pr−1 elements of one type and pr−1 of another, but the notation in (4.12) involves p^{h−2} for h = 1, which is not defined, and the displayed indices d are not explicitly bounded to the interval [j p^r, (j+1)p^r). The disjointness of the counted sets and their membership in D_{p^r,j} should be stated precisely; otherwise the conclusion #D_{p^r,j} ≥ p^r is not fully justified.
minor comments (4)
  1. [Throughout] There are numerous typos: 'monominal', 'corresponging', 'we btain', 'Herzong' in Remark 2.2, and inconsistent punctuation in equations. These should be corrected.
  2. [Remark 2.2(2)] The claim I = I_2(...) is presented as an outline; since I is non-prime in the main example, it would help to state explicitly how the Hilbert–Burch argument handles the torsion in Cl(X).
  3. [Section 4, equation (4.4)] The Euler characteristic formula χ(O_Y(−σjp^r C)|_{σp^r C}) = −2p^r is plausible, but the preceding displayed cases for O_Y(−nC)/O_Y(−(n+1)C) should be cross-checked: the middle case says O_{P^1}(−2) for n ≡ 1,6,8 mod 12, and the additive computation is not shown.
  4. [Section 4, [III-1]] In the p = 5 case, the formula for the index of 2○_d after taking p^{h−1}-th powers should be written consistently for h = 1; currently the exponent p^{h−2} is invalid for h = 1.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the p≥5 non-Noetherian proof is a standalone modular count; self-citations supply independent prior criteria, not fitted conclusions.

full rationale

The paper's central claim—R'_s(I) is Noetherian iff char(K) ∈ {2,3}—does not reduce to its inputs by construction. The char 2/3 half is an explicit verification of condition (A4): ξ^2 = 1 + x_{0,2} − (x_{1,1})^2 in F_4 and ξ^3 = 1 − x_{0,3} − (x_{1,1})^3 in F_6, using the precomputed decompositions F(m,ℓ)=A(m,ℓ)+B(m,ℓ); the conclusion is not hidden in the definitions. The p≥5 half is a direct counting argument: Lemma 4.1 shows C_{p^r,j}\D_{p^r,j} is a K-basis of H^1, Remark 4.2 derives H^0=0 from #D_{p^r,j}≥p^r together with the Euler characteristic (4.4), and the rest of Section 4 verifies the required congruences (4.10), (4.12), (4.22), (4.32), (4.45), (4.48), (4.66), (4.67) case-by-case from Lemma 4.3. No parameter is fitted to the target dichotomy; the modular coefficients are fixed binomial/rational numbers. The paper does rely on the author's prior papers [15] and [16] for the general finite-generation criteria (Theorem 1.1 and the (B2)–(B3) equivalence), and it extends them from the prime-ideal case to the non-prime I with the sentence 'We can prove the equivalence of them in the same way' without giving the proof. That is a missing-support/correctness risk, not a circular reduction: the cited results are published, parameter-free theorems whose assumptions do not include the target example, and the target example's arithmetic is new. The heavy binomial expansions in Section 4 (e.g., the nine-term combination before (4.64)) are asserted without intermediate algebra; a wrong coefficient would break (4.6), but that is a verifiability concern, not circularity. There is no equation here that is equivalent to its input by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The ledger is clean on parameters and entities: nothing is fitted and nothing is invented. The burden sits in axioms 1, 2, and 4: two previously published identifications/criteria (both largely the author's own) and one asserted inequality. The EMU condition is a combinatorial definition, not an invented entity.

assumptions (5)
  • standard math Cox(Y) is isomorphic to the extended symbolic Rees ring R'_s(I) (Remark 2.2(3), citing (2.8) in Kurano [16]).
    Published identification between the geometric object (Cox ring of the blow-up Y) and the algebraic object (extended symbolic Rees ring of the kernel ideal I). It is the author's own published theorem, used as a black box throughout; the whole Example 1.4 statement is phrased through it.
  • domain assumption The finite generation criteria of Inagawa-Kurano [15] (Theorems 3.1, 3.2, Proposition 5.9) and Kurano-Nishida [16] transfer verbatim from the prime-ideal case to the non-prime case.
    Theorems 1.1 and 1.2 are proved by saying the equivalence of (A0)-(A3) and (B2)-(B3) holds 'in the same way' as in [15], which treats prime I. The headline example requires the non-prime case: I is not prime since d = 24 and Cl(X) has torsion (see (2.2) and Section 4). If the transfer fails, both the char 2/3 and char >= 5 verdicts lose their engine. This is the largest unproved-in-text step.
  • standard math Cutkosky's theorem [2]: in positive characteristic, existence of a negative curve forces the symbolic Rees ring to be Noetherian; used for Theorem 1.3(1) ('One can prove (1) in the same way as Cutkosky [2]').
    Published theorem; supports the 0 < W < 1 part of Theorem 1.3, which is not used in the headline example (W = 1 there).
  • domain assumption h^0(F)_K depends only on the characteristic of K, and h^0(F)_{F_p} >= h^0(F)_Q for every prime p (Section 4, before Example 1.4(2)(i)).
    Asserted as 'easy to see' and 'easy to check'; used to prove Example 1.4(2)(i) that finite generation over Q implies finite generation over any K. Not used for the headline g = 13/6 dichotomy.
  • standard math Hilbert-Burch theorem and the Cohen-Macaulay computation behind the presentation of I as the ideal of 2-minors (Remark 2.2(2)).
    Standard commutative algebra background invoked to write I = I_2(matrix) and to identify its symbolic powers; the presentation is then verified by localizing at xyz.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Infinitely generated symbolic Rees rings of positive characteristic." pith.science (2026). https://pith.science/paper/RRFNMVTH

@misc{pith2026250804127,
  author       = {Pith},
  title        = {Pith review of: Infinitely generated symbolic Rees rings of positive characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RRFNMVTH}},
  note         = {Machine review of arXiv:2508.04127}
}
read the original abstract

Let X be a toric variety over a field K determined by a triangle. Let Y be the blow-up at (1,1) in X. In this paper we give some criteria for finite generation of the Cox ring of Y in the case where Y has a curve C such that C^2 \le 0 and C.E=1 (E is the exceptional divisor). The natural surjection Z^3 \rightarrow Cl(X) gives the ring homomorphism K[Z^3] \rightarrow K[Cl(X)]. We denote by I the kernel of the composite map K[x,y,z] \subset K[Z^3] \rightarrow K[Cl(X)]. Then Cox(Y) coincides with the extended symbolic Rees ring R's(I). In the case where Cl(X) is torsion-free, this ideal I is the defining ideal of a space monomial curve. Let Delta be the triangle (4.1) below. Then I is the ideal of K[x,y,z] generated by 2-minors of the 2*3-matrix {{x^7, y^2, z},{y^{11}, z, x^{10}}}. (In this case, there exists a curve C with C^2=0 and C.E=1. This ideal I is not a prime ideal.) Applying our criteria, we prove that R's(I) is Noetherian if and only if the characteristic of K is 2 or 3.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [15]

    Inagawa and K

    T. Inagawa and K. Kurano , Some necessary and sufficient condition for finite generation of symbolic Rees rings , J. Algebra 619 (2023), 153–198

    work page 2023

  2. [16]

    Kurano , Equations of negative curves of blow-ups of Ehrhart rings of rational convex polygons, J

    K. Kurano , Equations of negative curves of blow-ups of Ehrhart rings of rational convex polygons, J. Algebra 590 (2022), 413-438

    work page 2022

  3. [1]

    R. C. Cowsik , Symbolic powers and number of defining equations , Algebra and its applica- tions (New Delhi, 1981), Lecture Notes in Pure and Appl. Math. 91, D ekker, New York, 1984, 13-14

    work page 1981

  4. [2]

    S. D. Cutkosky , Symbolic algebras of monomial primes , J. reine angew. Math. 416 (1991), 71-89

    work page 1991

  5. [3]

    Cutkosky and K

    S D. Cutkosky and K. Kurano , Asymptotic regularity of powers of ideals of points in a weighted projective plane , Kyoto J. Math. 51 (2011), 25–45

    work page 2011

  6. [4]

    Ebina , Master theses, Meiji University 2017 (Japanese)

    T. Ebina , Master theses, Meiji University 2017 (Japanese)

    work page 2017

  7. [5]

    J. L. Gonz ´alez and K. Karu , Some non-finitely generated Cox rings , Compos. Math. 152 (2016), 984–996

    work page 2016

  8. [6]

    Gonz ´alez-Anaya, J

    J. Gonz ´alez-Anaya, J. L. Gonz ´alez and K. Karu , Constructing non-Mori Dream Spaces from negative curves , J, Algebra 539 (2019), 118–137

    work page 2019

Show all 23 references
  1. [7]

    Gonz ´alez-Anaya, J

    J. Gonz ´alez-Anaya, J. L. Gonz ´alez and K. Karu , Curves generating extremal rays in blowups ofweighted projective planes , J. of London Math. Soc. 104 (2021), 1342–1362

    2021

  2. [8]

    Gonz ´alez-Anaya, J

    J. Gonz ´alez-Anaya, J. L. Gonz ´alez and K. Karu , The geography of negative curves , Michigan Math. J. 75 (2025), no. 1, 89-118

    2025

  3. [9]

    Gonz ´alez-Anaya, J

    J. Gonz ´alez-Anaya, J. L. Gonz ´alez and K. Karu , Nonexistence of negative curves , Int. Math. Res. Not. IMRN 2023, no. 16, 14368-14400

    2023

  4. [10]

    S. Goto, K. Nishida and K.-i. W atanabe , Non-Cohen-Macaulay symbolic blow-ups for space monomial curves and counterexamples to Cowsik’s ques tion, Proc. Amer. Math. Soc. 120 (1994), 383–392

    1994

  5. [11]

    Hartshorne , Algebraic Geometry, GTM 52, Springer Verlag, 1977

    R. Hartshorne , Algebraic Geometry, GTM 52, Springer Verlag, 1977

    1977

  6. [12]

    He , New examples and non-examples of Mori dream spaces when blowing up toric surfaces; arXiv:1703.00819, 2017

    Z. He , New examples and non-examples of Mori dream spaces when blowing up toric surfaces; arXiv:1703.00819, 2017

    2017 arXiv

  7. [13]

    Herzog, Generators and relations of Abelian semigroups and semigro up rings, Manuscripta Math

    J. Herzog, Generators and relations of Abelian semigroups and semigro up rings, Manuscripta Math. 3 (1970), 175–193

    1970

  8. [14]

    Huneke , Hilbert functions and symbolic powers , Michigan Math

    C. Huneke , Hilbert functions and symbolic powers , Michigan Math. J. 34 (1987), 293–318

    1987

  9. [17]

    Kurano and N

    K. Kurano and N. Matsuoka , On finite generation of symbolic Rees rings of space mono- mial curves and existence of negative curves , J. Algebra 322 (2009), 3268-3290

    2009

  10. [18]

    Kurano and K

    K. Kurano and K. Nishida , Infinitely generated symbolic Rees rings of space monomial curves having negative curves , Michigan Math. J. 68 (2019), 405–445

    2019

  11. [19]

    Matsuura , Master theses, Meiji University 2019 (Japanese)

    M. Matsuura , Master theses, Meiji University 2019 (Japanese). 42 KAZUHIKO KURANO

    2019

  12. [20]

    Nagata, On the 14-th Problem of Hilbert , Amer

    M. Nagata, On the 14-th Problem of Hilbert , Amer. J. Math. 81 (1959) 766-772

    1959

  13. [21]

    Sannai and H

    A. Sannai and H. Tanaka , Infinitely generated symbolic Rees algebras over finite field s, Algebra and Number theory, 13 (2019), 1879–1891

    2019

  14. [22]

    Totaro , Moving codimension-one subvarieties over finite fields , Amer

    B. Totaro , Moving codimension-one subvarieties over finite fields , Amer. J. Math. 131 (2009), 18151833

    2009

  15. [23]

    Uchisawa, Master theses, Meiji University 2017 (Japanese)

    K. Uchisawa, Master theses, Meiji University 2017 (Japanese). Kazuhiko Kurano Department of Mathematics Faculty of Science and Technology Meiji University Higashimita 1-1-1, Tama-ku Kawasaki 214-8571, Japan kurano@meiji.ac.jp http://www.isc.meiji.ac.jp/~kurano

    2017

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.