Hilbert-Kunz Multiplicity of Fibers and Bertini Theorems

Austyn Simpson; Rankeya Datta

REVIEW 3 major objections 3 minor 46 references

Hilbert-Kunz Multiplicity of Fibers and Bertini Theorems

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that a general hyperplane section preserves a Hilbert–Kunz multiplicity bound for equidimensional schemes in prime characteristic, without any normality assumption.

desk verdict Genuine generalization of CRST17, but the proof leans on an unproved in-preparation uniform bound that is load-bearing; worth refereeing, not desk-rejecting. read the letter →

arxiv 1908.04819 v2 pith:LXTM47SA submitted 2019-08-13 math.AG math.AC

Rankeya Datta , Austyn Simpson This is my paper · ORCID

classification math.AGmath.AC MSC 13A3514B0514E2514N05

keywords Hilbert–KunzmultiplicityBertinitheoremprimecharacteristichyperplanesectionsuniformboundsrelativeFrobeniusF-finiteringsequidimensionalschemes

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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The 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.

The 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.

The 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.

Overall: 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.

Who 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.

Referee Report

3 major / 3 minor

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.

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 (3)

[Section 3.1, Theorem 3.1.7 and inequality (3.8)] 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 4.1, Theorem 4.1.2(4)] 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 3.2, Theorem 3.2.2] 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.

minor comments (3)

[Section 4, proof of Theorem 4.1(2)] There is a typo in 'deﬁntion' in the proof of Theorem 4.1(2); the manuscript should be proofread for such typographical errors.
[Section 3.1, Definition 3.1] 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 4.1, Theorem 4.1.2(3)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Bertini theorem is derived from an external uniform-bound input, and no fitted parameter or definitional equivalence is renamed as a prediction.

full rationale

The paper's derivation chain is not circular. The target result is the Bertini-type statement for the Hilbert-Kunz property P_HK,λ; this property is an input hypothesis, not the output of a fit. The proof verifies axioms (A1) and (A2') for P_HK,λ using the Cumino-Greco-Manaresi framework. (A1) comes from Kunz's theorem, and (A2') is verified via the uniform estimates in Section 3. The main uniformity step, inequality (3.8), invokes Theorem 3.1.7, attributed to [PTY], which is an external in-preparation result and not authored by Datta or Simpson. This is a genuine source of correctness risk: if Theorem 3.1.7 is unavailable, the uniform bounds UBPH-K and the verification of (A2') collapse. But that is an unproved-input gap, not circularity, and the hard rules require an exhibited reduction by construction to flag a circular step. The only self-citation is [DM19, Lemma 4.8] (Datta-Murayama), used in Theorem 4.1.2(4) to factor a finitely generated field extension into a finite purely inseparable extension followed by a finitely generated separable extension. That is a standard, independently checkable field-theoretic fact and is not load-bearing for the main Bertini theorem; it appears in an auxiliary part of the proof and does not smuggle in the Hilbert-Kunz conclusion. No fitted constants are renamed as predictions, no uniqueness theorem by the same authors is invoked to forbid alternatives, and the known result of Carvajal-Rojas-Schwede-Tucker is explicitly generalized rather than relabeled. Therefore the paper exhibits no significant circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No numerical parameters are fitted to data: lambda is a fixed input threshold and e0, d0, C are existential constants in the uniform estimates. The central claim rests on standard results in commutative algebra plus two load-bearing external inputs that are not yet public: [PTY, Theorem 3.1.7] and [DM19, Lemma 4.8]. No new objects, forces, or mediators are introduced.

assumptions (7)

domain assumption Theorem 3.1.7 [PTY]: for A, R, M as in Setting 3.1, a single constant C uniformly bounds the Hilbert-Kunz lengths over all regular κ(p)-algebras.
Invoked in the proof of Theorem 3.1.8 to bound the cokernels in equation (3.8) and thereby obtain UBPH-K; the cited source is listed only as 'In preparation', so the bound functions as an unverified black box.
domain assumption For every finitely generated field extension L/κ(p) there is a Hasse diagram with an intermediate finite purely inseparable extension followed by a finitely generated separable extension ([DM19, Lemma 4.8]).
Used in Theorem 4.1.2(4) to spread PHK,λ from purely inseparable extensions to arbitrary finitely generated field extensions; the cited source is a preprint not yet published.
standard math Direct Summand Theorem: a regular ring in characteristic p is a direct summand of every module-finite extension ring.
Used in Theorem 3.1.8 to conclude A[t1,...,tδ] to R splits, yielding module-finite split fibers over polynomial subrings.
standard math CGM86 axiomatic Bertini theorem: a local property satisfying (A1) and (A2') yields Bertini statements for finite type morphisms with separably generated residue field extensions (Theorem 4.2).
The whole strategy applies this published framework to the property PHK,λ.
standard math Smirnov's upper semi-continuity of Hilbert-Kunz multiplicity on locally equidimensional rings of finite type over a field (Theorem 4.6).
Used to prove axiom (A3) and the openness of eHK(X;λ) in Corollary 4.7.
standard math Kunz's theorem: a flat local map with regular closed fiber preserves Hilbert-Kunz multiplicity (Theorem 4.5).
Used to verify axiom (A1) for PHK,λ.
standard math Monsky's existence and polynomial growth of the Hilbert-Kunz function for Noetherian local rings in characteristic p (Definition-Theorem 2.2.1).
Provides the foundational definition and asymptotic expansion of Hilbert-Kunz multiplicity used throughout the paper.

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Pith. "Pith review of Hilbert-Kunz Multiplicity of Fibers and Bertini Theorems." pith.science (2026). https://pith.science/paper/LXTM47SA

@misc{pith2026190804819,
  author       = {Pith},
  title        = {Pith review of: Hilbert-Kunz Multiplicity of Fibers and Bertini Theorems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LXTM47SA}},
  note         = {Machine review of arXiv:1908.04819}
}

abstract

Let $k$ be an algebraically closed field of characteristic $p > 0$. We show that if $X\subseteq\mathbb{P}^n_k$ is an equidimensional subscheme with Hilbert--Kunz multiplicity less than $\lambda$ at all points $x\in X$, then for a general hyperplane $H\subseteq\mathbb{P}^n_k$, the Hilbert--Kunz multiplicity of $X\cap H$ is less than $\lambda$ at all points $x\in X\cap H$. This answers a conjecture and generalizes a result of Carvajal-Rojas, Schwede and Tucker, whose conclusion is the same as ours when $X\subseteq\mathbb{P}^n_k$ is normal. In the process, we substantially generalize certain uniform estimates on Hilbert--Kunz multiplicities of fibers of maps obtained by the aforementioned authors that should be of independent interest.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

I. M. Aberbach and F. Enescu, Lower bounds for Hilbert--Kunz multiplicities in local rings of fixed dimension , Mich. Math. J. 57 (2008), pp. 1--16

work page 2008

[2] [2]

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work page 1993

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M. Blickle and F. Enescu, On rings with small Hilbert--Kunz multiplicity , Proc. Amer. Math. Soc. 132 (2004), pp. 2505--1509

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