{"id":"10940804-ce3e-45b7-9a5f-e7ceff9ef480","arxiv_id":"1908.04819","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For equidimensional projective schemes over an algebraically closed field of characteristic p, having Hilbert-Kunz multiplicity below lambda everywhere is preserved by taking general hyperplane sections.","lead":"This paper proves a Bertini-type theorem: if a space built from polynomial equations in positive characteristic has Hilbert-Kunz multiplicity below a threshold at every point, then a general hyperplane slice does too. It removes a normality assumption from an earlier result and settles a conjecture of Carvajal-Rojas, Schwede, and Tucker.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorem rests on an unproved, in-preparation uniform bound (Theorem 3.1.7, [PTY]) used at (3.8); without it, UBPH-K and the verification of (A2') collapse.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing concern: Theorem 3.1.7 from the in-preparation paper [PTY] is used essentially at (3.8) to control cokernels uniformly, and no proof or public version is provided. My read confirms that this is the single most serious gap. The rest of the argument is internally coherent: the reduction to geometrically reduced fibers, the use of relative Frobenius, the axiomatic framework of Cumino-Greco-Manaresi, and the equidimensionality arguments all appear consistent. The concern is not a detected error but a missing foundational input, so the appropriate verdict is CONDITIONAL rather than ACCEPT or REJECT. The paper should either supply a proof of the needed special case of Theorem 3.1.7 in an appendix or wait for [PTY] to become publicly available.","tokens_in":34814,"tokens_out":45186,"duration_ms":403740,"concrete_test":"Independently prove the special case of Theorem 3.1.7 used at (3.8): for A a regular F-finite domain, R module-finite over A[t_1,...,t_δ], and M_i obtained from Lemma 3.1.5, establish directly that a single constant C satisfies ℓ_{R_{κ(p)^{1/p^d},x}}((M_i⊗ κ(p)^{1/p^d})_x / P^{[pe]}(...)) ≤ C p^{e dim((M_i⊗...)_x)} for all p∈Spec A_g, d≥0, and x, using only published tools such as [PT18, Lemma 3.5], [Tuc12, Lemma 3.3], and [Smi16]. If such a proof cannot be completed without new uniformity input, or if a computational search over small p, e, and d for a family with unbounded ratios succeeds, the central argument is conditional on [PTY].","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The load-bearing step is the application of Theorem 3.1.7 at inequality (3.8) in the proof of Theorem 3.1.8. Theorem 3.1.7 asserts that for a Noetherian ring A of characteristic p, a finitely generated A-algebra R, and a finitely generated R-module M, a single constant C bounds the Hilbert-Kunz length ℓ_{R_{Γ,P}}((M_Γ)_P/P^{[pe]}(M_Γ)_P) by C p^{e dim((M_Γ)_P)} uniformly over all primes p∈Spec A, all regular κ(p)-algebras Γ, all P∈Spec(R_Γ), and all e≥1. The cited source [PTY] is marked 'In preparation', and no proof or public version is supplied. In the proof, this theorem is the only source of uniformity in p and d; it controls the cokernels ψ_{i,e} in (3.8), which yields the uniform convergence estimate (3.10) and hence Theorem 3.1.8, Corollary 3.1.9, Theorem 3.2.2, and finally the verification of (A2') in Theorem 4.1.2(3). Although the proof only needs the special case Γ=κ(p)^{1/p^d}, that special case is not proved in the paper either. If Theorem 3.1.7 fails, or if its constant cannot be chosen independent of the auxiliary regular κ(p)-algebra Γ, the UBPH-K estimates and the Bertini theorem for arbitrary equidimensional X are unsupported. The paper contains no internal contradiction; the gap is a missing foundational input.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a Bertini-type theorem for Hilbert--Kunz multiplicity: if X is an equidimensional finite type subscheme of projective n-space over an algebraically closed field of characteristic p>0 and e_HK(O_{X,x}) < lambda for all x in X, then the same bound holds for a general hyperplane section. The proof follows the Cumino--Greco--Manaresi axiomatic framework, verifying axioms (A1), (A2'), and (A3) for the property e_HK < lambda. The main technical contribution is a series of uniform estimates for Hilbert--Kunz lengths of fibers of finite type maps, including cases with geometrically reduced and non-reduced fibers. This generalizes prior work of Carvajal-Rojas--Schwede--Tucker, which required normality.","tokens_in":35189,"tokens_out":4699,"duration_ms":51705,"significance":"If correct, the main theorem settles a conjecture of Carvajal-Rojas, Schwede, and Tucker and substantially broadens the known Bertini statements for Hilbert--Kunz multiplicity to arbitrary equidimensional schemes. The uniform fiber estimates in Section 3 are of independent interest and are stated in considerable generality. The paper is well structured and gives detailed proofs of the new reductions, including the treatment of nilpotent fibers via Frobenius twisting and untwisting. However, the central uniform bound on which the argument rests is quoted from an unpublished, in-preparation source, so the completeness of the proof cannot currently be verified. I found no internal contradiction in the portions that are proved in the manuscript.","major_comments":[{"comment":"The uniform estimate that drives the entire proof is quoted from [PTY], marked 'In preparation,' and no proof is supplied. At inequality (3.8) this theorem is the only input that bounds the cokernels of the maps psi_{i,e} uniformly in the base prime, the auxiliary regular algebra, and the localizing point; without it the uniform convergence bound (3.10) is unsupported, and consequently Theorem 3.1.8, Theorem 3.2.2, and the verification of (A2') in Theorem 4.1.2(3) collapse. Since the application in this paper only needs the special case Gamma = kappa(p)^{1/p^d}, the authors should either prove that special case in the present paper or supply a public, citable proof of Theorem 3.1.7.","section":"Section 3.1, Theorem 3.1.7 and inequality (3.8)"},{"comment":"Part (4) of Theorem 4.1.2 depends on [DM19, Lemma 4.8], cited as a preprint, to decompose an arbitrary finitely generated field extension L/kappa(p) into a purely inseparable extension followed by a separable extension. This decomposition is load-bearing for the very-general version in Theorem 4.1(3). The authors should include a proof of the lemma or replace the reference with a published source, since the current manuscript does not allow a referee to verify that step.","section":"Section 4.1, Theorem 4.1.2(4)"},{"comment":"Theorem 3.2.2, used in the proof of Theorem 4.1.2(3), inherits the dependence on Theorem 3.1.7 through Theorem 3.1.8 and Corollary 3.1.9. This is not an additional mathematical error, but it means the unresolved status of Theorem 3.1.7 propagates directly to the main Bertini theorem; the authors should state this dependence explicitly when revising.","section":"Section 3.2, Theorem 3.2.2"}],"minor_comments":[{"comment":"There is a typo in 'defintion' in the proof of Theorem 4.1(2); the manuscript should be proofread for such typographical errors.","section":"Section 4, proof of Theorem 4.1(2)"},{"comment":"In Definition 3.1 the notation e_HK(F_{kappa(p)^{1/p^d},x}) is used for what is later explained to be the Hilbert--Kunz multiplicity of the module with respect to the maximal ideal; this should be clarified at first use to avoid confusion with the local-ring Hilbert--Kunz multiplicity.","section":"Section 3.1, Definition 3.1"},{"comment":"The step in the proof of part (3) applying Theorem 2.2.2 uses the fact that R_L,y -> R_{kappa(p)^{1/p^d},x} is faithfully flat; this is true for field extensions, but the justification could be stated more explicitly.","section":"Section 4.1, Theorem 4.1.2(3)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is carefully written and the main idea appears sound, but acceptance cannot be recommended until the proof of Theorem 3.1.7 is either included or made available in a public, citable form. The dependence on an in-preparation reference is not a minor presentation issue; it is a load-bearing gap that prevents verification of the main theorem. The second external dependence, on [DM19], is less severe but should likewise be resolved. I would ask the authors to supply the missing proofs or, failing that, to state the main theorem as conditional on the uniform bound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: this is a genuine result, not a repackaging. The authors prove a Bertini theorem for Hilbert-Kunz multiplicity that removes the normality hypothesis from CRST17, and they do it with a new proof that handles non-reduced and non-normal fibers via relative Frobenius and twisting by A^{1/p^e}. The uniform estimates in Section 3 are also new and go beyond CRST17.\n\nThe paper is well organized. The strategy is the CGM axiomatic framework, and the hard part is verifying (A2') for the property eHK<λ. They do this through a local theorem (4.1.2) that shows the property spreads from the generic fiber to nearby fibers. The proof of Theorem 3.1.8 is the technical heart, and it is clever: they reduce to geometrically reduced fibers, use the splitting of Noether normalization, and control the cokernel of Frobenius comparisons. The handling of nilpotents via Lemma 3.2.3 and the twisting argument is careful.\n\nThe soft spot is real and it is load-bearing. Theorem 3.1.7, cited as [PTY] 'In preparation', supplies the uniform bound on Hilbert-Kunz lengths over all primes of the base, all regular κ(p)-algebras Γ, and all points of the fiber. This is exactly what makes the cokernel control uniform in (3.8). There is no proof of it in the paper, and no public version of [PTY] is available. Even the special case Γ=κ(p)^{1/p^d} that the proof actually uses is not proved here. If that constant cannot be chosen independent of Γ, the UBPH-K estimates collapse, and with them the verification of (A2') and the Bertini theorem for arbitrary equidimensional X. This is not an internal contradiction; the argument is coherent modulo that input. But it is a missing foundational block.\n\nThe second citation, [DM19, Lemma 4.8] for the Hasse diagram in part (4) of Theorem 4.1.2, is a preprint, but it's a standard-looking field-extensions fact and much less concerning.\n\nOverall: the paper deserves a serious referee. The main theorem is significant and likely correct, but the referee should verify the status of [PTY] and ask the authors to include a proof of the special case they need, or at least make the preprint available. I'd send it out; I'd also tell the authors that the paper should not be published in its current form while the main estimate is in-preparation elsewhere.\n\nWho is this for: anyone working on F-singularities, Bertini theorems, or uniform behavior of Hilbert-Kunz multiplicity. I would cite it once the dependence is resolved.","headline":"Genuine generalization of CRST17, but the proof leans on an unproved in-preparation uniform bound that is load-bearing; worth refereeing, not desk-rejecting.","tokens_in":35728,"tokens_out":2370,"would_cite":true,"duration_ms":23423,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13A35","14B05","14E25","14N05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a general hyperplane section preserves a Hilbert–Kunz multiplicity bound for equidimensional schemes in prime characteristic, without any normality assumption.","keywords":["Hilbert–Kunz multiplicity","Bertini theorem","prime characteristic","hyperplane sections","uniform bounds","relative Frobenius","F-finite rings","equidimensional schemes"],"falsifier":"Take k algebraically closed of characteristic p, choose an equidimensional non-normal subscheme X⊆P^n_k with eHK(O_{X,x})<λ for all x, pick a general hyperplane H, and compute eHK(O_{X∩H,y}) at each point y of the section; if any value is ≥λ, Theorem 4.1(1) is false. A concrete test would be a non-normal surface in $P^{3}$ whose hyperplane section curve has a point of Hilbert–Kunz multiplicity exceeding the ambient λ, verifiable by computer algebra on small examples.","tokens_in":34620,"feed_emoji":"✂️","tokens_out":5976,"duration_ms":55909,"temperature":0.7,"pith_summary":"The paper proves a Bertini-type theorem for Hilbert–Kunz multiplicity: if X is an equidimensional subscheme of projective space over an algebraically closed field of prime characteristic, and every local ring of X has Hilbert–Kunz multiplicity below a fixed bound λ, then a general hyperplane section X∩H also has every local Hilbert–Kunz multiplicity below λ. This extends a known result that required X to be normal, and answers a conjecture posed in that earlier work. The proof works by establishing uniform control of Hilbert–Kunz multiplicities across the fibers of a finite-type map, a property the authors call UBPH-K, and then feeding this into a standard axiomatic framework for Bertini theorems. A reader should care because Hilbert–Kunz multiplicity measures singularity severity in prime characteristic, and the result says that hyperplane sections do not make singularities worse in this measure, even for non-normal schemes.","feed_headline":"General hyperplanes preserve Hilbert–Kunz bounds","feed_subtitle":"An equidimensional variety's Hilbert–Kunz multiplicity below λ survives a general hyperplane cut, without normality.","key_machinery":"The load-bearing mechanism is the Uniform Boundedness Property of Hilbert–Kunz (UBPH-K): for a finite-type map A→R of F-finite rings with equidimensional generic fibers, constants e0,d0,C and a nonzero g∈A control, uniformly over all primes p of A_g, all purely inseparable base changes κ(p)^{1/p^d}, and all points of the fiber, the difference between the local Hilbert–Kunz multiplicity and its e-th truncation. The proof establishes UBPH-K by first treating geometrically reduced fibers, using the relative Frobenius and a comparison of Frobenius pushforwards, then removing the reducedness hypothesis by twisting with a large Frobenius power and untwisting. This uniform estimate verifies axiom (A2′) of the axiomatic Bertini framework, which supplies the general-hyperplane conclusion.","core_discovery":"The central discovery is Theorem 4.1: over an algebraically closed field k of characteristic p>0, if X⊆P^n_k is equidimensional and eHK(O_{X,x})<λ for all x∈X, then for a general hyperplane H⊆P^n_k, eHK(O_{X∩H,y})<λ for all y∈X∩H. The same conclusion holds for finite-type images with separably generated residue field extensions. For closed embeddings, the open locus where the bound holds is preserved under intersection with a general hyperplane (version (2)); if k is uncountable and the bound is ≤λ, a very general hyperplane also preserves it (version (3)). The proof reduces the statement to verifying two axioms for the property eHK<λ: a flat-with-regular-fibers base-change axiom, already known, and a spreading-out axiom that the paper proves using uniform estimates on Hilbert–Kunz multiplicities of fibers of finite-type maps.","pith_inferences":["Beyond the paper: if the cited in-preparation uniform bound [PTY] becomes publicly available, the proof would likely extend to arbitrary bases or to non-equidimensional generic fibers.","Beyond the paper: the uniform fiber estimates may also imply Bertini-type statements for other Frobenius invariants such as F-signature, without the normality hypothesis that earlier normal-case results required.","Beyond the paper: the very general version (3) relies on uncountability of the ground field; for countable algebraically closed fields the conclusion might fail or require a different argument, offering a concrete test case.","Beyond the paper: the twisting-by-Frobenius method used to reduce reducedness could be adapted to prove uniform bounds for test ideals or F-thresholds in families."],"forward_implications":["A general hyperplane section of an equidimensional singular scheme in characteristic p has Hilbert–Kunz multiplicity bounded by the same λ, so cutting down dimension does not worsen this singularity measure.","The result holds for finite-type images with separably generated residue fields, not only for closed embeddings.","For closed embeddings, the open locus where eHK<λ is stable under intersecting with a general hyperplane.","Over uncountable fields, the ≤λ version survives for very general hyperplanes.","The uniform fiber estimates in Section 3 are of independent interest for studying Hilbert–Kunz multiplicity in families."],"supporting_citations":[{"why":"Establishes existence and basic properties of Hilbert–Kunz multiplicity, the central invariant of the paper.","marker":"[Mon83]"},{"why":"Provides the flat-with-regular-fibers equality used for axiom (A1) and the p-degree formulas used throughout the uniform estimates.","marker":"[Kun76]"},{"why":"Contains the conjecture and the normal-case theorem that this paper generalizes.","marker":"[CRST17]"},{"why":"Supplies the axiomatic Bertini framework connecting axioms (A1) and (A2′) to general-hyperplane conclusions.","marker":"[CGM86]"},{"why":"Quoted without proof as the source of the uniform bound Theorem 3.1.7 over all primes and regular algebras, on which the UBPH-K estimates rest.","marker":"[PTY]"},{"why":"Provides upper semi-continuity of Hilbert–Kunz multiplicity, used to verify axiom (A3) and the openness of the locus eHK<λ.","marker":"[Smi16]"},{"why":"Gives the global Hilbert–Kunz multiplicity formulation used in Lemma 2.3.1 and in the proof of the uniform estimates.","marker":"[DSPY19]"},{"why":"Characterizes flat maps with geometrically reduced fibers via purity of the relative Frobenius, used in the geometrically reduced case of Theorem 3.1.8.","marker":"[Dum95]"}],"fun_headline_variants":["Hyperplane cuts preserve Hilbert–Kunz thresholds","Bertini for Hilbert–Kunz: no normality needed","Hilbert–Kunz bound survives general hyperplanes","Hilbert–Kunz bound survives general cuts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on a uniform bound, quoted from an in-preparation work [PTY], asserting that a single constant controls Hilbert–Kunz lengths over every prime of the base, every regular algebra over each residue field, and every point of the fiber; if that bound fails, the paper's uniform estimates and hence the Bertini theorem collapse.","fun_headline_variants_meta":{"raw":{"variants":["Hyperplane cuts preserve Hilbert–Kunz thresholds","Bertini for Hilbert–Kunz: no normality needed","Hilbert–Kunz bound survives general hyperplanes","Hilbert–Kunz bound survives general cuts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000786,"raw_usage":{"total_tokens":3446,"prompt_tokens":901,"completion_tokens":2545,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":517,"completion_tokens_details":{"reasoning_tokens":2477}},"tokens_in":517,"tokens_out":2545,"duration_ms":27776,"temperature":1.0,"reasoning_tokens":2477,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:33:03.528187+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take k algebraically closed of characteristic p, choose an equidimensional non-normal subscheme X⊆P^n_k with eHK(O_{X,x})<λ for all x, pick a general hyperplane H, and compute eHK(O_{X∩H,y}) at each point y of the section; if any value is ≥λ, Theorem 4.1(1) is false. A concrete test would be a non-normal surface in $P^{3}$ whose hyperplane section curve has a point of Hilbert–Kunz multiplicity exceeding the ambient λ, verifiable by computer algebra on small examples.","supporting_citations":[],"review_version":1}