Frobenius liftable hypersurfaces

Jakub Witaszek; Supravat Sarkar; Tatsuro Kawakami

REVIEW 3 major objections 4 minor 1 cited by

Frobenius liftable hypersurfaces

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that every reduced divisor in projective space over an algebraically closed field of positive characteristic that admits a Frobenius lift modulo p^2 is toric, and derives a characteristic-zero corollary for toric covers.

desk verdict Genuinely new classification result with a clean proof idea, but the reduction from reduced divisors to components in Theorem A needs a justification before the result is fully established. read the letter →

arxiv 2507.12198 v1 pith:CMVB5MYK submitted 2025-07-16 math.AG

Tatsuro Kawakami , Supravat Sarkar , Jakub Witaszek This is my paper · ORCID

classification math.AG MSC 13A3514M25

keywords FrobeniusliftabletoricdivisorprojectivespacelogBottvanishingtotallyinvariantdivisorspositivecharacteristichyperplanearrangementsPicardrankone

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

Frobenius liftability is a characteristic-$p$ analogue of having a polarized endomorphism, and this paper asks how rigid it is for divisors in projective space. The main theorem says that if a reduced divisor $D\subset\mathbb{P}^n_k$ admits a Frobenius lift modulo $p^2$, then $D$ is a toric divisor: after a linear automorphism of $\mathbb{P}^n$, it is a union of coordinate hyperplanes. Equivalently, the irreducible components of $D$ are hyperplanes whose defining equations are linearly independent. In characteristic zero the paper derives a corollary: a smooth projective variety of Picard rank one that admits a finite surjective morphism from a toric pair with toric preimage divisor must itself be projective space, with $D$ toric. This is the projective-space case of the general expectation that Frobenius liftability is a toric condition, and it gives evidence for the conjecture that totally invariant divisors of endomorphisms of $\mathbb{P}^n$ are linear.

What carries the argument

The mechanism is the sheaf of logarithmic $1$-forms $\Omega^1_X(\log D)$ together with the subsheaf $(\Omega^1_X(\log D))^\xi$ of sections fixed by the Frobenius-linear map $\xi\colon F_*\Omega^1_X(\log D)\to\Omega^1_X(\log D)$ induced by the Frobenius lift; this invariant subsheaf is called the magic cover in the cited work. Because $\mathbb{P}^n$ is simply connected, a locally constant $\mathbb{F}_p$-sheaf on the normal-crossing locus is constant, so a nonzero section over a general line, which deforms uniquely inside the invariant subsheaf, patches to a global section of $\Omega^1_X(\log D)$. The contradiction comes from log Bott vanishing, which forces $H^0(X,\Omega^1_X(\log D))=H^0(X,\Omega^1_X)=0$ for $X=\mathbb{P}^n$ while a divisor of degree at least two supplies a nonzero section on a general line.

What would settle it

Try to construct a Frobenius lift modulo $p^2$ of $(\mathbb{P}^2_k, C)$ where $C$ is a smooth conic, for example $C=V(xy+yz+zx)$ in characteristic $2$; the theorem predicts no such lift exists. A more targeted check is to compute $H^1(\mathbb{P}^2, \Omega^1_{\mathbb{P}^2}(\log C)(-C+A))$ for an ample $A$ on any candidate lift and see whether the log Bott vanishing of Proposition 3.5 is violated.

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Extended reading notes

Core claim

The central claim, stated as the main theorem, is a complete classification: for an algebraically closed field $k$ of characteristic $p>0$ and a reduced divisor $D$ on $\mathbb{P}^n_k$, if the pair $(\mathbb{P}^n_k,D)$ is Frobenius liftable modulo $p^2$, then $D$ is a toric divisor with respect to the standard toric structure on $\mathbb{P}^n_k$, up to an automorphism of $\mathbb{P}^n_k$. The proof isolates the case of a prime divisor and shows that its degree must be one, so the only possible components are hyperplanes. A final linear-algebra step shows the defining equations of these hyperplanes must be linearly independent, because a dependent arrangement would violate the log-canonicity that Frobenius liftability forces. As a corollary, the paper shows that over the complex numbers a smooth projective variety of Picard rank one admitting a finite surjective morphism from a toric pair with toric preimage divisor is itself projective space, and the divisor is toric.

Load-bearing premise

The classification rests on the imported statement that every Frobenius-liftable pair satisfies a certain vanishing of cohomology of logarithmic differential forms twisted by an ample line bundle; if that vanishing fails for the exact definition of Frobenius liftability used here, the proof collapses.

Editorial extensions

If this is right

Every reduced Frobenius-liftable divisor in $\mathbb{P}^n_k$ is a hyperplane arrangement whose defining linear forms are linearly independent.
There are no Frobenius-liftable smooth hypersurfaces of degree at least $2$ in $\mathbb{P}^n$.
For complex varieties of Picard rank one, a finite surjective toric cover with toric preimage divisor forces the base to be $\mathbb{P}^n$ and the divisor to be toric.
The conjecture that totally invariant divisors of endomorphisms of $\mathbb{P}^n_{\mathbb{C}}$ are linear is confirmed for endomorphisms arising from unramified Frobenius lifts.

Reading between the lines

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

The deformation mechanism suggests a testable converse on other simply connected Fano varieties: if log Bott vanishing holds and a covering family of rational curves can replace lines, Frobenius liftability of a prime divisor may force the divisor to be linear.
Theorem A makes Frobenius liftability equivalent to toricity for reduced divisors in projective space, so a computational search for Frobenius lifts modulo $p^2$ could serve as a toricity test for explicit divisors.
The classification would extend if the imported log Bott vanishing holds under the weaker Frobenius-liftability condition used elsewhere in the literature; checking that split injection is the natural next step.

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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

The paper proves a genuinely new statement: an F-liftable reduced divisor on projective space must be a toric divisor. The proof uses the magic cover to deform a line-restricted logarithmic form into a global section, and the strategy is well executed. If correct, Theorem A and its characteristic-zero corollary Theorem B are substantial additions to the Frobenius-liftable analogue of the totally-invariant-divisor problem, and they upgrade the Occhetta-Wiśniewski result in a meaningful way.

What the paper does well: the main contradiction mechanism is explicit, the dimension-counting lemma (3.9) is clean, and the final discrepancy argument is elegant. The debt to [AWZ21] is honestly acknowledged, and the use of [AWZ21] for the split injection in Proposition 3.5 does not worry me. The authors explicitly tightened Definition 2.2 to match the hypotheses of [AWZ21], so the stress-test concern about that imported result does not land: the cited work is peer-reviewed and the definition is tailored to it.

The soft spot is elsewhere. In the proof of Theorem A, the authors write that for an irreducible component D' of D, the pair (X,D') is F-liftable, and therefore deg(D')=1 by Theorem 3.10. This is not automatic under the stricter Definition 2.2. The normal-crossing locus of (X,D') is generally larger than the normal-crossing locus of (X,D); a lift eU of the smaller locus does not obviously extend to a lift of the larger one with the required relative normal-crossing property for eD'. The paper gives no argument here. If this inheritance fails, Theorem A only establishes the prime-divisor case, and the reduced divisor case is left open. This needs a lemma or a revised argument.

There is also a citation risk: Theorem 3.11 depends on [KT24] (a preprint) and [Kaw22] (to appear), both overlapping with the authors. That is not a flaw in itself, but it means part of the load-bearing structure is not yet widely vetted. The spreading-out argument in Theorem B is standard.

For a reader in F-singularity theory or toric geometry, this paper is worth the time. The main idea is strong and the overall direction is clearly correct, but the component-wise reduction is a genuine gap that a referee should ask to be fixed. Send it to peer review: the result deserves referee time, and the missing justification is exactly what peer review should catch.

Referee Report

3 major / 4 minor

Summary. The paper studies reduced divisors D on P^n_k in characteristic p>0 that are Frobenius liftable modulo p^2 in the sense of Definition 2.2. It proves Theorem A: any such D is a toric divisor, i.e., after an automorphism of P^n_k, a union of coordinate hyperplanes with linearly independent defining equations. The proof combines log Bott vanishing for F-liftable pairs (Proposition 3.5), a computation of logarithmic 1-forms (Theorem 3.6), and a deformation argument using the 'magic cover' of ξ-invariant logarithmic forms (Theorem 3.10) to show every irreducible component has degree one, then uses log canonicity to rule out dependent hyperplanes. Theorem B derives a characteristic-zero consequence: if a smooth projective variety X of Picard rank one admits a finite surjective morphism f:Y→X such that (Y,f^{-1}(D)_red) is a toric pair, then X≅P^n and D is a toric divisor.

Significance. If Theorem A is correct, it gives a complete classification of reduced Frobenius-liftable divisors in projective space: they are exactly hyperplane arrangements with linearly independent equations. This is a strong positive-characteristic analogue of the totally invariant divisor conjecture and connects F-liftability to toric geometry. The proof strategy is elegant, combining the log Bott vanishing framework of Achinger–Witaszek–Zdanowicz with the magic cover and a line-deformation argument that uses semicontinuity to force constancy. The paper is concise and the main statements are crisp. However, the central proof rests on imported results whose hypotheses are not verified in the manuscript, and a few steps in the applications need explicit justification. The result is significant but, as written, not fully self-contained.

major comments (3)

[Section 3.1, Proposition 3.5] The proof of Proposition 3.5 reduces to a split surjection F_*Ω^{d-i}_U(log D)→Ω^{d-i}_U(log D) on the normal crossing locus U, imported from [AWZ21, Proposition 3.2 and Variant 3.2.2]. This proposition is load-bearing: it supplies the vanishing H^1(X,Ω^{[1]}_X(log D))=0 in Theorem 3.6 and the vanishing used in Theorem 3.10. The manuscript does not state the precise variant being cited, nor does it verify that the restriction of a pair satisfying Definition 2.2 to its normal crossing locus U satisfies the hypotheses of that variant. In particular, the role of the condition that eD|eU be normal crossing relative to W2 should be made explicit. Please add this verification or state and prove the needed split surjection.
[Proof of Theorem A, first sentence] The assertion that each irreducible component D' of D is F-liftable is not justified. Definition 2.2 imposes a normal-crossing condition on the lift of the normal crossing locus of the pair, and this locus can be larger for (X,D') than for (X,D); the same lift of Frobenius does not automatically make eD' normal crossing relative to W2 on the larger lifted locus. Since Theorem 3.10 is stated and proved only for a prime divisor that is F-liftable as a pair, this step needs an argument.
[Section 3.2, Theorem 3.11] The descent from a toric pair (Y,D_Y) to (X,D) being F-liftable is by [KT24, Theorem 3.11], but Definition 2.2 is explicitly stronger than [KT24, Definition 2.2]. Please confirm that the lift produced by [KT24, Theorem 3.11] satisfies the normal-crossing-on-the-lift condition of Definition 2.2, or give a direct construction of such a lift. As written, Theorem A may not apply to the output of [KT24, Theorem 3.11], and the same concern affects the proof of Theorem B.

minor comments (4)

[Lemma 2.4] The phrase 'rank p^{dim X}' is confusing: by Lemma 2.1, the fixed points of a Frobenius-linear bijection on a rank-n vector space form an F_p-vector space of dimension n, so the rank should be dim X unless 'rank' is intended to mean the cardinality of stalks; please clarify.
[Proposition 3.5, final sentence] The sentence 'we have H^j(Ω^{[i]}_X(log D)(p^e A - D))' is missing the conclusion '= 0'; please correct the typo.
[Theorem 3.10] The notation for U0, U, G, and the maps φ and π is hard to follow; in particular, the displayed definition of G appears to conflate pushforward and pullback. Please rewrite this paragraph with distinct symbols for the two projections and define G as a subsheaf of the pullback of the magic cover.
[Proof of Theorem 3.6] The sentence 'From the above exact sequence, we can deduce the assertions' is correct only after using the assumption H^1(X,Ω^1_X)≠0 to force δ_X to be nonzero and hence surjective; this step should be spelled out for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: F-liftability does not define toricity, and the cited [AWZ21] splitting is an independent theorem, not a renamed conclusion.

full rationale

Walking the derivation chain, Definition 2.2 defines F-liftability by the existence of a flat W2-lift eX, closed subschemes eDr, and a Frobenius lift eF with eF*(eDr|eU)=p(eDr|eU); it never mentions hyperplanes, linear equations, or toric divisors. Theorem A's conclusion that D is a toric divisor up to automorphism is therefore not contained in the hypothesis by definition. The main reduction is Proposition 3.5, which derives log Bott vanishing from F-liftability; its proof is a citation to [AWZ21, Proposition 3.2 and Variant 3.2.2] for a split surjection of logarithmic forms. This citation is load-bearing and comes from a paper sharing an author, so it is self-citation, but the cited proposition is a published, parameter-free theorem whose hypotheses are the F-liftability hypotheses and whose conclusion is a vanishing statement; it does not assume toricity or the target classification. The subsequent argument—Lemma 3.2 residue sequence, Theorem 3.6, Lemma 3.8 line-degree computation, Theorem 3.10 deformation of sections of the magic cover, and the discrepancy contradiction in Theorem A—is a new and nontrivial derivation. The paper itself flags that Definition 2.2 is stricter than [KT24, Definition 2.2] and says the extra normal-crossing-locus condition is needed for the deformation-theoretic arguments of [AWZ21]; this is an omitted verification or potential gap if the cited variant does not apply, but a gap is not circularity. No equation is used as its own conclusion, no fitted parameter is renamed a prediction, and the cited theorems do not reduce to the result they are used to prove. Hence no significant circularity.

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

No free parameters or invented entities; the paper is a pure proof. The main imported inputs are prior Frobenius-liftability results from the same research program (AWZ21, AWZ23, Kaw22, KT24, KT25a), so the theorem rests on those theorems being correct.

assumptions (5)

domain assumption The etale sheaf of xi-invariant logarithmic 1-forms is constructible and locally constant of rank p^n over the isomorphism locus (Lemma 2.4).
Imported from [AWZ21] as the magic cover; the deformation argument in Theorem 3.10 needs its stalks to have constant F_p-rank along the family of lines. Section 2.2, Lemma 2.4.
domain assumption Log Bott vanishing for F-liftable pairs: H^j(X, Omega^{[i]}_X(log D)(-D+A)) = 0 for ample A.
Used to get H^1(log D)=0 in Theorem 3.6 and H^0(log D)=H^0(Omega^1) in Theorem 3.10. Proof cites [AWZ21, Prop. 3.2 and Variant 3.2.2] rather than deriving the split injection. Section 3.1, Proposition 3.5.
standard math Any locally constant etale sheaf on V = X minus a closed set of codimension at least 3 is constant because projective space is simply connected.
Zariski-Nagata purity invoked in Theorem 3.10 to conclude the invariant subsheaf extends and is constant. Section 3.1, around the deformation of lines.
standard math A finite injective Frobenius-linear endomorphism of a finite-dimensional vector space over an algebraically closed field is bijective and its fixed locus is F_p^r (Lemma 2.1).
Justifies the equality between fixed-section dimension and h^0 on each line. Proof via [Sta, Tag 0A3L].
domain assumption F-liftability descends to closed fibers through [KT24, Theorem 3.11], and toric pairs are F-liftable via [Kaw22, Remark 2.7(3)].
Theorem 3.11 and Theorem B rely on these results from related preprints for spreading out and descent from toric pairs. These are not proved in this paper.

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Pith. "Pith review of Frobenius liftable hypersurfaces." pith.science (2026). https://pith.science/paper/CMVB5MYK

@misc{pith2026250712198,
  author       = {Pith},
  title        = {Pith review of: Frobenius liftable hypersurfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CMVB5MYK}},
  note         = {Machine review of arXiv:2507.12198}
}

abstract

Let $D$ be a reduced divisor in $\mathbb P^n_k$ for an algebraically closed field $k$ of positive characteristic $p > 0$. We prove that if $(\mathbb P^n_k, D)$ is Frobenius liftable modulo $p^2$, then $D$ is a toric divisor. As a corollary, we show that if there exists a finite surjective morphism $f\colon Y\to X$ onto a smooth projective complex variety $X$ of Picard rank $1$ such that $(Y, f^{-1}(D)_{\mathrm{red}})$ is a toric pair, then $X$ is the projective space and $D$ is a toric divisor.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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math.AG 2025-06 conditional novelty 5.0 of 10

A smooth projective weak Fano threefold of Picard rank 2 that is either a toric image or admits an int-amplified endomorphism is toric, with three explicit non-Fano exceptions.

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Works this paper leans on

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Pith tools

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

[1] [1]

Piotr Achinger, Jakub Witaszek, and Maciej Zdanowicz, Global F robenius liftability I , J. Eur. Math. Soc. (JEMS) 23 (2021), no. 8, 2601--2648. 4269423

work page 2021

[2] [2]

, Global F robenius liftability II : surfaces and F ano threefolds , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 24 (2023), no. 1, 329--366. 4587749

work page 2023

[3] [3]

Arnaud Beauville, Endomorphisms of hypersurfaces and other manifolds, Internat. Math. Res. Notices (2001), no. 1, 53--58. 1809497

work page 2001

[4] [4]

39, Cambridge University Press, Cambridge, 1993

Winfried Bruns and J\" u rgen Herzog, Cohen- M acaulay rings , Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. 1251956

work page 1993

[5] [5]

Brian Conrad, From normal crossings to strict normal crossings, 2016

work page 2016

[6] [6]

Pierre Deligne and Luc Illusie, Rel\`evements modulo p^2 et d\'ecomposition du complexe de de R ham , Invent. Math. 89 (1987), no. 2, 247--270. 894379

work page 1987

[7] [7]

Dolgachev, Logarithmic sheaves attached to arrangements of hyperplanes, J

Igor V. Dolgachev, Logarithmic sheaves attached to arrangements of hyperplanes, J. Math. Kyoto Univ. 47 (2007), no. 1, 35--64. 2359100

work page 2007

[8] [8]

52, Springer Science & Business Media, 2013

Robin Hartshorne, Algebraic geometry, vol. 52, Springer Science & Business Media, 2013

work page 2013