{"id":"b106335e-7c21-4775-9f64-e8d26d747582","arxiv_id":"2508.04127","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the ideal I generated by the 2-minors of the matrix {{x^7, y^2, z}, {y^11, z, x^10}}, the extended symbolic Rees ring is Noetherian if and only if the field has characteristic 2 or 3.","lead":"This paper finds a concrete algebraic object (the symbolic Rees ring attached to a toric surface blow-up) that is finitely generated only when the base field has characteristic 2 or 3, and infinitely generated in every other prime characteristic. It matters because it sharpens what positive characteristic can do to Cox rings, which are central to the theory of Mori dream spaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Non-Noetherian half rests on unverified binomial congruences; a single wrong coefficient in Section 4 linear combinations would break (4.6) and the p≥5 dichotomy.","rationale":"The central claim is the characteristic dichotomy for the non-prime ideal I. The Noetherian half for p=2,3 is a short computation and appears sound. The load-bearing half is non-Noetherian for p≥5. The proof reduces to a purely combinatorial assertion (4.6) proven by modular binomial expansions. These expansions are numerous and unverified; the risk is real because the theorem's conclusion is a universal statement over all primes, and one exceptional prime (e.g., p=44777) is singled out, showing the authors themselves found a case where generic remainders vanish. This legitimate concern is the one the reader flagged, though the specific [III-2] accusation appears mistaken. Since the paper otherwise has a coherent geometric structure and the p=2,3 cases are checkable, the appropriate verdict remains conditional pending independent verification of the Section 4 congruences. No change to the reader's verdict is needed.","tokens_in":34698,"tokens_out":16755,"duration_ms":175237,"concrete_test":"Implement the definitions of F, A, B for triangle (4.1) in a CAS (Sage/Macaulay2) and directly test the membership of x_{10d+5,12d+6} (or x_{10d+7,12d+8}) in A+B for the congruences (4.10),(4.12),(4.22),(4.32),(4.45),(4.48),(4.66),(4.67) for small primes p=5,7,11,13,17,19,23,29,31,37,41,43,47 and r=1,2, j=1,2,...,p. Also symbolically verify the big linear combination in [III-5] (and analogues) by re-expanding with Lemma 4.3 and checking that all unwanted components cancel; if any coefficient fails, the p≥5 dichotomy is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the p≥5 non-Noetherian argument, the goal is (4.3), which is reduced to the counting bound (4.6) #D_{p^r,j}≥p^r. The bound is proved case-by-case for p≡1,3,7,9 mod 10 and p=5 by showing that certain elements 2○_d or 3○_d lie in D_{p^r,j}. Each case hinges on a congruence of the form x_{10d+5,12d+6} or x_{10d+7,12d+8} ≡0 modulo A+B, derived from (4.9) and Lemma 4.3 after a long binomial expansion. The expansions are mostly stated without derivation; the most extreme is [III-5], where a nine-term linear combination of (4.54)–(4.62) is asserted to cancel all components except q1/q2, with no intermediate algebra. A single incorrect binomial coefficient or combination coefficient would make the claimed membership false; then H^0 need not vanish and (C3) could hold, so the 'infinitely generated for every p≥5' assertion would fail. I did not find a definite error by hand; the reader's specific claim that [III-2] mis-states p=10f+1 is not confirmed by the text (that section assumes p=10f+3 and relies on 3f(3f+2)/2≠0, which is correct). The concern is therefore the unverified computational burden, not a known gap. Secondary: Theorem 1.1's transfer from the prime-ideal case to the non-prime ideal of Example 1.4 is asserted by 'in the same way' with no written derivation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":34867,"tokens_out":3994,"duration_ms":55877,"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":[{"comment":"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.","section":"Section 3, proof of Theorem 1.1"},{"comment":"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.","section":"Section 4, [III-3], proof of (4.23)"},{"comment":"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.","section":"Section 4, [III-5], proof of (4.50)–(4.51)"},{"comment":"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.","section":"Section 4, counting and equations (4.12), (4.22), (4.66), (4.67)"}],"minor_comments":[{"comment":"There are numerous typos: 'monominal', 'corresponging', 'we btain', 'Herzong' in Remark 2.2, and inconsistent punctuation in equations. These should be corrected.","section":"Throughout"},{"comment":"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).","section":"Remark 2.2(2)"},{"comment":"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.","section":"Section 4, equation (4.4)"},{"comment":"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.","section":"Section 4, [III-1]"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the author's own previous papers [15] and [16], and the transfer of the finite-generation criteria to the non-prime case is asserted rather than proved. This is not itself a problem, but it is a central point that the editor should ask the author to address. The large computational part of Section 4 appears coherent, but it is not reproducible from the text in its current form; a referee with access to a computer algebra system could verify the stated reductions quickly. If no verification is provided, the p ≥ 5 non-Noetherian claim should not be considered fully established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plainly: this is a serious paper. The main example gives the first complete characteristic dichotomy for an extended symbolic Rees ring of a non-prime ideal in the W=1 toric setting: for the ideal of 2-minors of the displayed matrix, the ring is Noetherian exactly in characteristics 2 and 3. If the Section 4 arithmetic holds, that is a real result, the kind people will cite.\n\nWhat is actually new: Theorem 1.3(2), which pins finite generation to the order of O_Y(σp^r C) restricted to σp^r C being a p-power, plus the (C0)-(C4) equivalence. The proof structure is sound. The char 2 and char 3 cases in Section 4 are short, and I verified the factorizations and the set data in (4.2); those are done cleanly.\n\nThe soft spot is the p≥5 half. The non-Noetherian claim is reduced to counting the set D_{p^r,j}, and the architecture of Lemma 4.1, the Euler characteristic -2p^r, and the dimension count is coherent. But between (4.10) and (4.67) there is a long chain of binomial congruences, many asserted without intermediate algebra. The stress-test note correctly identifies [III-5] as the vertex: a nine-term linear combination with a single printed coefficient, and the membership of 2○_d or 3○_d in D_{p^r,j} depends on those coefficients being exactly right. I did not find a definite error, and the reader's specific accusation about [III-2] misstating the residue class does not hold up: that section assumes p=10f+3 and the coefficient 3f(3f+2)/2 is indeed nonzero mod p. But the difference between 'no error found' and 'verified' matters here. The author should supply the detailed expansion or ship a computer check; otherwise the referee has to redo a lot of arithmetic by hand.\n\nThe second soft spot is Theorem 1.1. It transfers Inagawa-Kurano's criteria from prime I to non-prime I by 'in the same way', but the headline example is non-prime, so that transfer is load-bearing. It may be straightforward for the author, but it is not written out.\n\nThe paper is for commutative algebraists and algebraic geometers working on symbolic Rees rings, Cox rings, and Mori dream spaces. It is on a live problem, and the example is concrete enough to justify the effort. I would send it to a knowledgeable referee with a request to check the char p≥5 congruences; that is the difference between accept and reject. I would not desk-reject it.","headline":"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.","tokens_in":35679,"tokens_out":3555,"would_cite":true,"duration_ms":40065,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13A30","14M25","14C20","13A35"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["symbolic Rees rings","Cox rings","toric varieties","positive characteristic","Mori dream spaces","finite generation","space monomial curves","blow-ups"],"falsifier":"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.","tokens_in":34302,"feed_emoji":"🧮","tokens_out":12521,"duration_ms":142764,"temperature":0.7,"pith_summary":"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.","feed_headline":"Symbolic Rees ring: Noetherian only for p=2 and 3","feed_subtitle":"An explicit non-prime ideal flips to infinitely generated for every prime characteristic p ≥ 5.","key_machinery":"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)","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the F/A/B/ξ framework and the (A4)/(B2) criteria that Section 3 extends to non-prime ideals.","marker":"[15]"},{"why":"Identifies Cox(Y) with R'_s(I) and fixes the blow-up and curve notation used throughout.","marker":"[16]"},{"why":"Gives the geometric interpretation of symbolic Rees rings as Cox rings and supplies the positive-characteristic criterion that Theorem 1.3 adapts.","marker":"[2]"},{"why":"Cited as the model for proving Theorem 1.3(2) via nef non-semi-ample divisors.","marker":"[22]"},{"why":"Used in Remark 2.2(2) to write I as the ideal of 2-minors of the displayed 2×3 matrix.","marker":"[13]"}],"fun_headline_variants":["Symbolic Rees ring: Noetherian only for p=2,3","Infinitely generated symbolic Rees rings for p≥5","Toric blow-up: Cox ring fails finite generation for p≥5","Characteristic p: Noetherian only when p=2 or 3","Cox ring: infinite generation unless p=2 or p=3"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Symbolic Rees ring: Noetherian only for p=2,3","Infinitely generated symbolic Rees rings for p≥5","Toric blow-up: Cox ring fails finite generation for p≥5","Characteristic p: Noetherian only when p=2 or 3","Cox ring: infinite generation unless p=2 or p=3"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000415,"raw_usage":{"total_tokens":2048,"prompt_tokens":878,"completion_tokens":1170,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":622,"completion_tokens_details":{"reasoning_tokens":1076}},"tokens_in":622,"tokens_out":1170,"duration_ms":11357,"temperature":1.0,"reasoning_tokens":1076,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T00:57:56.210731+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Inagawa and K","cited_arxiv_id":null,"evidence_quote":"Supplies the F/A/B/ξ framework and the (A4)/(B2) criteria that Section 3 extends to non-prime ideals."},{"cited_title":"Kurano , Equations of negative curves of blow-ups of Ehrhart rings of rational convex polygons, J","cited_arxiv_id":null,"evidence_quote":"Identifies Cox(Y) with R'_s(I) and fixes the blow-up and curve notation used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the geometric interpretation of symbolic Rees rings as Cox rings and supplies the positive-characteristic criterion that Theorem 1.3 adapts."},{"cited_title":"Totaro , Moving codimension-one subvarieties over finite fields , Amer","cited_arxiv_id":null,"evidence_quote":"Cited as the model for proving Theorem 1.3(2) via nef non-semi-ample divisors."},{"cited_title":"Herzog, Generators and relations of Abelian semigroups and semigro up rings, Manuscripta Math","cited_arxiv_id":null,"evidence_quote":"Used in Remark 2.2(2) to write I as the ideal of 2-minors of the displayed 2×3 matrix."}],"review_version":1}