On the canonical bundle formula and effective birationality for Fano varieties in char $p>0$

Xintong Jiang

REVIEW 4 major objections 7 minor 40 references

On the canonical bundle formula and effective birationality for Fano varieties in char $p>0$

T0 review · 4 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper establishes effective birationality for strongly F-regular weak Fano varieties in characteristic p>0: from dimension and Gorenstein index alone, fixed n,m give vol(-nK_X)>(2d)^d and a birational |-mK_X|.

desk verdict New results on normalization and the threefold canonical bundle formula are worth taking seriously, but the effective birationality theorem is not proved as written: it rests on an unproved bounded-covering statement and the induction step has a numerical gap. read the letter →

arxiv 2501.12041 v3 pith:2ELPUFD2 submitted 2025-01-21 math.AG

Xintong Jiang This is my paper · ORCID

classification math.AG MSC 14E3014J4514G1713A35

keywords FanovarietiespositivecharacteristiceffectivebirationalityF-singularitiesstrongF-regularcanonicalbundleformulaboundednessBABconjecture

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 works in characteristic $p>0$, where the standard boundedness tools from characteristic $0$ fail because vanishing theorems break down. Its central result is an effective birationality statement for weak Fano varieties with good F-singularities: if $X$ is strongly F-regular, of fixed dimension $d$, and has Gorenstein index $I$ not divisible by $p$, then there are integers $n$ and $m$, depending only on $d$ and $I$, such that $\operatorname{vol}(-nK_X)>(2d)^d$ and the linear system $|-mK_X|$ defines a birational map. This gives a positive-characteristic analogue of the effective birationality predicted by the BAB conjecture, restricted to the F-regular setting. Along the way the paper proves that boundedness is stable under normalization, and it establishes a usable canonical bundle formula for threefold Fano-type fibrations over bases of positive dimension in large characteristic, together with adjunction formulas on non-klt centers.

What carries the argument

The load-bearing mechanism is the interplay between F-pure centers and F-adjunction. An F-pure center is a closed subvariety where a pair $(X,\Delta)$ is sharply F-pure but not strongly F-regular; it is, roughly, the minimal locus where the Frobenius splitting degenerates. For such a center $W$, F-adjunction gives a canonical divisor relation $(K_W+\Delta_W)=(K_X+\Delta)|_W$, with the Cartier index of $K_W+\Delta_W$ controlled by that of $K_X+\Delta$. The paper's engine is Remark 4.1, which asserts that an ample $\mathbb{Q}$-divisor $D$ with $\operatorname{vol}(D)\ge(2d)^d$ yields, for any two general points $x,y$, a boundary $\Delta\sim_{\mathbb{Q}} D$ that is F-pure at $x$ with a unique F-pure center from a fixed finite family covering $X$ and not F-regular at $y$; this lets the argument induct on the center's dimension via Lemma 4.3, which cuts the center down by adding an ample divisor whose restriction to the center has degree $>k^k$.

What would settle it

To settle the central claim, one could look for a strongly F-regular weak Fano variety $X$ of dimension $d$ with $IK_X$ Cartier and $p\nmid I$ such that the minimal $m$ with $|-mK_X|$ birational is not bounded in terms of $d$ and $I$, or such that $\operatorname{vol}(-nK_X)\le(2d)^d$ for all $n$ in the range allowed by the proof. More directly, one can test the unproved assertion of Remark 4.1: exhibit an ample $\mathbb{Q}$-divisor $D$ on some $X$ with $\operatorname{vol}(D)\ge(2d)^d$ and two general points $x,y$ for which every $\Delta\sim_{\mathbb{Q}}D$ either fails to be F-pure at $x$ with a unique F-pure center in the claimed finite family, or is F-regular at $y$.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 4.4: for fixed natural numbers $d$ and $I$ with $p\nmid I$, there exist integers $n$ and $m$ depending only on $d$ and $I$ such that every strongly F-regular weak Fano variety $X$ of dimension $d$ with $IK_X$ Cartier satisfies $\operatorname{vol}(-nK_X)>(2d)^d$ and $|-mK_X|$ gives a birational map. The proof produces, for each such $X$, a bounded covering family of F-pure centers and then runs an induction on the dimension of these centers: an ample-divisor cutting lemma shrinks the F-pure center while preserving sharp F-purity at one general point and non-F-regularity at a second, until the center is a point, at which point an F-potentially birational divisor theorem forces the relevant anti-canonical linear system to separate points. A corollary replaces the Gorenstein-index hypothesis by a lower bound on the F-signature, giving $m$ depending only on the dimension and the F-signature bound.

Load-bearing premise

The argument rests on an unproved assertion, Remark 4.1, that from any ample $\mathbb{Q}$-divisor $D$ with $\operatorname{vol}(D)\ge(2d)^d$ one can always find, for any two general points $x$ and $y$, a boundary $\Delta\sim_{\mathbb{Q}} D$ whose only F-pure center containing $x$ is a subvariety from a fixed finite family covering $X$, while the pair is not F-regular at $y$; if that assertion fails, the effective birationality conclusion is unsupported.

Editorial extensions

If this is right

For fixed $d$ and $I$ with $p\nmid I$, all strongly F-regular weak Fano $d$-folds with $IK_X$ Cartier share a single anti-canonical linear system $|-mK_X|$ that is birational, so their images in projective space have bounded degree.
Corollary 4.5: if instead of fixing the Gorenstein index one assumes $X$ is $\epsilon$-strongly F-regular, the same birationality holds with $m$ depending only on $d$ and $\epsilon$, because a uniform multiple of $K_X$ is Cartier.
Since volumes and Gorenstein indices are bounded for bounded families, the theorem gives a new boundedness statement: the set of strongly F-regular weak Fano varieties with fixed dimension and Gorenstein index not divisible by $p$ is birationally bounded.
The canonical bundle formula of Theorem 3.3 implies that a contraction of a threefold of Fano type is again of Fano type in large characteristic when the general fibers are normal and a uniform multiple of the relative canonical divisor is Cartier.
Theorem 2.4 shows boundedness is stable under normalization in arbitrary characteristic, so future boundedness arguments may assume varieties are normal without loss.

Reading between the lines

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

Because the proof only needs sharp F-purity and F-adjunction, a natural extension would be to replace strong F-regularity by klt or lc with a uniform Cartier index in large characteristic; if Remark 4.1 can be proved in that setting, effective birationality would follow for $\epsilon$-lc weak Fano varieties, closely matching the BAB prediction.
The dependence of the canonical bundle formula on a characteristic threshold $p_0=p_0(\Phi)$ suggests that any counterexample to effective birationality in char $p$ would have to live in small characteristic relative to the boundary coefficients or the Gorenstein index; a systematic search over small $p$ for Fano threefolds with unbounded $m$ would test this.
The normalization-stability theorem removes a known obstruction to reducing boundedness questions to normal varieties in char $p$, so one could combine it with the canonical bundle formula to attempt a full BAB-type boundedness statement for $\epsilon$-lc threefolds by first normalizing and then applying the induction on F-pure centers; the author does not carry out this step.
The volume threshold $(2d)^d$ and the cutting condition $H^k\cdot Z>k^k$ appear to be the exact analogues of the characteristic-0 thresholds, suggesting the same effective-birationality constants could hold uniformly in all characteristics once the missing assertion is supplied.

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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

Two things you should know about this one. It has two genuinely new pieces: Theorem 2.4 (boundedness is stable under normalization in char p) and Theorem 3.3 (a canonical bundle formula for threefold Fano fibrations in large characteristic). And it has a headline theorem, Theorem 4.4 (effective birationality for strongly F-regular weak Fano varieties), which is not proved as written. I would not send the effective birationality to the bank yet, but the other results deserve a serious look.

Theorem 2.4 is a clean observation: boundedness of a family of projective varieties in char p survives normalization. The Frobenius-base-change argument and the conductor stabilization are credible, and the statement is useful. Theorem 3.3 is the real substance. For a threefold klt Fano-type fibration to a surface or curve, it gives a canonical bundle formula for p above a bound depending on the coefficients or the Gorenstein index. The proof is compressed and leans on Witaszek, Benozzo, and Bernasconi-Martin, but the strategy—control the geometric generic fiber, kill inseparable base change with a large-p bound, then invoke known CBF results—is coherent. I did not find a fatal gap there, though a referee will want the abbreviated steps expanded.

Now the soft spot. Theorem 4.4 depends on Remark 4.1, an unproved assertion that any ample divisor with volume at least (2d)^d yields a bounded covering family of F-pure centers with the separation property. In characteristic 0 this is a deep theorem; in char p, no such boundedness result for F-pure centers is established, and the cited [HMX14, 7.1] is a characteristic-0 ACC statement. So the engine is missing. On a second pass I also see a numerical problem in the induction: Lemma 4.3 gives a new boundary (1−δ)Δ_i + cA with c<1 and A∼_Q −l n K_{X_i}, whose class is roughly −(n+1+c l n)K_{X_i}; equating that to the stated replacement −4 l n K_{X_i} forces c≈4 for l≥2, contradicting c<1. The induction step is not closed. And the theorem is stated for every dimension, while Theorem 4.1 assumes resolution of singularities, which is only known up to dimension 3 in char p. A missing reference: the proof cites Theorem 3.8 for F-adjunction but the paper only has Theorem 3.7; minor.

Net: the author is honest, knows the literature, and the paper is not circular. The canonical bundle formula part is valuable. But Theorem 4.4, as written, is unsupported. If I were the editor, I would send it to a serious referee—there is enough real content to justify the time—and expect a major revision that either proves the covering-family assertion or restricts the theorem to dimensions where resolution is known and the numeric issues can be repaired. I would not cite the effective birationality in its current form.

Referee Report

4 major / 7 minor

Summary. This paper studies birational geometry in positive characteristic. It proves that boundedness is stable under normalizations (Theorem 2.4), establishes a canonical bundle formula for klt threefold Fano fibrations over positive-dimensional base in sufficiently large characteristic (Theorem 3.3), applies this to non-klt center adjunction (Theorems 3.4-3.5) and to preservation of Fano type under contractions (Corollary 3.6), and proves an effective birationality statement for strongly F-regular weak Fano varieties with bounded Gorenstein index (Theorem 4.4), with a corollary for epsilon-strongly F-regular varieties (Corollary 4.5). The main emphasis is Theorem 4.4, which aims at a positive-characteristic analogue of Birkar's effective birationality in the F-regular, bounded-index setting.

Significance. If Theorem 4.4 were established, it would be a significant step toward effective birationality and BAB-type boundedness in positive characteristic. The earlier sections also contain useful tools: Theorem 2.4 is a clean characteristic-free boundedness observation, Theorem 3.3 is a usable canonical bundle formula for threefold Fano fibrations, and Corollaries 1.4-1.5 are likely to be useful for complement constructions. The paper is not circular: it derives results from external theorems such as the three-dimensional MMP, F-adjunction, and boundedness of geometrically integral del Pezzo surfaces. However, the central Theorem 4.4 is currently a conditional statement, because the covering-family engine in Remark 4.1 is asserted rather than proved, and the induction in its proof has a concrete numerical inconsistency.

major comments (4)

[Section 4, Remark 4.1] The assertion that an ample Q-divisor D with vol(D) >= (2d)^d gives a bounded covering family of F-pure centers is the engine of Theorem 4.4, but it is not proved. The preceding paragraph only explains why potential birationality does not automatically become F-potential birationality, and the sentence 'combine the statements above and [HMX14, 7.1] and some tie breaking arguments' is not a proof. In characteristic 0 the analogous statement is a deep theorem of Birkar and depends on ACC for log canonical thresholds; [HMX14] is a characteristic-0 result and does not directly control F-pure centers in positive characteristic. Moreover, the phrase 'bounded covering family' is not defined in the paper, and the uniformity over a sequence of varieties is unclear. Since Theorem 4.4 invokes this statement for a sequence of varieties of arbitrary dimension, the main theorem is currently conditional on an unproved assertion.
[Section 4, proof of Theorem 4.4] The induction step contains a numerical inconsistency. With Delta_i ~Q -(n+1)K_{X_i} and A ~Q H = -lnK_{X_i}, Lemma 4.3 produces (1-delta)Delta_i + cA with 0<c<1, whose divisor class is -((1-delta)(n+1)+cln)K_{X_i}. The proof then asserts a replacement Delta'_i ~Q -4lnK_{X_i}. Equating these classes forces c = 4 - (1-delta)(n+1)/(ln), which for l>=1 and large n is at least about 3, contradicting c<1. Thus the stated application of Lemma 4.3 does not produce the claimed boundary, and the dimension-decreasing induction is not closed as written.
[Section 4, Theorem 4.4 and Definition 4.1] Theorem 4.4 is stated for every dimension d, but its proof uses F-potential birationality and Theorem 4.1, whose definition requires X to admit a resolution of singularities. In positive characteristic, resolution of singularities is not known beyond dimension 3. The k=0 branch of the proof therefore does not apply to the stated generality. Either the theorem should be restricted to d<=3, or a resolution-free argument (for example via alterations) must be supplied.
[Section 4, proof of Theorem 4.4, boundedness of l] The assertion that 'l is bounded' after F-adjunction is too terse. Here l_i is the smallest integer with vol(-lnK_{X_i}|_{G_i}) > k^k, and a uniform bound requires a lower bound on vol(-K_{X_i}|_{G_i}) for all i. The proof mentions that IK_X|_G is Cartier but does not state the volume lower bound that would follow from this. This is probably repairable, but it should be written out explicitly.

minor comments (7)

[Section 3, Theorem 3.3(1)] In the proof, the bound p0 >= 2/min(Phi) is used, but if 0 is in Phi the minimum is 0 and p0 is undefined. The statement should either require Phi to be contained in (0,1] or the proof should discard zero coefficients.
[Section 4, proof of Theorem 4.4] The reference to 'Theorem 3.8' is incorrect: the F-adjunction statement used here is Theorem 3.7. Please correct the cross-reference.
[Section 1, Corollary 1.4] The introduction refers to 'Theorem 3.5, 3.6', while the body numbers the relevant statements as Theorem 3.4 and Theorem 3.5. The numbering should be harmonized.
[Section 2, Theorem 2.4] The statement has a typo: 'a family P of projective varieties k' should read 'over k'. The proof of the positive-characteristic part is dense; in particular, the key openness assertion for geometric normality should be stated explicitly with a precise reference.
[Section 4, Corollary 4.5] The application of [BCRG+17, 4.8] gives that [1/epsilon+1]!K_X is Cartier, which may be divisible by p. Since the hypothesis is that the Gorenstein index is not divisible by p, the proof should explicitly pass to the actual index, which divides this factorial, rather than applying Theorem 4.4 with the factorial itself.
[Section 2, Remark 2.2 and references] There are small textual issues: 'can be defined defined' should be 'can be defined', and the reference [Hot22] spells the author's name 'Hotchster' instead of 'Hochster'.
[Section 4, Theorem 4.1] The notation T^0(Z,D_Z) is used without definition. It appears to be the intersection of images of trace maps over finite covers, but it should be defined explicitly before use.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's main claims are not derived from their own conclusions; the proof has unproved and numerically inconsistent steps, but those are correctness gaps, not circular reductions.

full rationale

The derivation chain is not circular. Theorem 4.4 is built from external inputs: the volume computation from nef-and-big Cartier divisors, F-adjunction from Schwede [Sch09], the MMP and boundedness results of Birkar, Bernasconi-Martin, Witaszek, Benozzo, and the F-singularity machinery of Blickle-Schwede-Tucker and Patakfalvi-Schwede-Zhang. No parameter is fitted to the target birationality statement, and no self-citation is load-bearing; indeed the paper contains no self-citations, since the author's own prior work is not invoked. The main weakness is not circularity but incompleteness: Remark 4.1 in Section 4 asserts, without proof, the bounded covering family of F-pure centers that powers the induction in Theorem 4.4, and the induction step replacing Delta_i ~_Q -(n+1)K_X by Delta'_i ~_Q -4lnK_X via Lemma 4.3 appears numerically incompatible with the lemma's requirement 0<c<1. These are proof gaps, not reductions of the claimed result to its own inputs, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on no fitted parameters and introduce no new mathematical entities. They rely on a portfolio of standard theorems (MMP, F-adjunction, boundedness of del Pezzo surfaces) plus one unproved assertion (Remark 4.1) and an implicit resolution assumption for arbitrary dimension.

assumptions (6)

standard math Existence of MMP for three-dimensional klt pairs over bases in characteristic p > 5.
Invoked as Theorem 2.2 to run minimal model programs in the proof of Theorem 3.3.
standard math Resolution of singularities for three-dimensional varieties in positive characteristic.
Cited as Theorem 2.1 and used to reduce to smooth models in the proof of Theorem 3.3.
standard math ACC for log canonical thresholds in dimension 2.
Used in Theorem 3.4 to ensure the set of discriminant coefficients is DCC.
standard math Boundedness of geometrically integral del Pezzo surfaces over Spec Z (Bernasconi-Martin).
Used in the dim Z=1 case of Theorem 3.3 to bound the generic fiber (X_eta,B_eta).
ad hoc to paper Existence of a bounded covering family of F-pure centers as stated in Remark 4.1.
This is an unproved assertion introduced by the paper and used as the main input for the induction in Theorem 4.4.
domain assumption Resolution of singularities (or alteration substitute) for strongly F-regular Fano varieties of arbitrary dimension in positive characteristic.
The definition of F-potentially birational in Theorem 4.1 requires X to admit a resolution; Theorem 4.4 is stated for all d, where resolution is unknown.

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Pith. "Pith review of On the canonical bundle formula and effective birationality for Fano varieties in char $p>0$." pith.science (2026). https://pith.science/paper/2ELPUFD2

@misc{pith2026250112041,
  author       = {Pith},
  title        = {Pith review of: On the canonical bundle formula and effective birationality for Fano varieties in char $p>0$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2ELPUFD2}},
  note         = {Machine review of arXiv:2501.12041}
}

read the original abstract

In this paper, we give some results on the birational geometry of varieties of Fano type and boundedness problems in positive characteristic, including a result ensuring that boundedness is invariant under normalizations, a canonical bundle formula for fibrations of Fano type which is easier to use, and the effective birationality of certain weak Fano varieties with good singularities, which is predicted by the BAB conjecture.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Positivity of the Moduli Part

Florin Ambro , Paolo Cascini , Vyacheslav Shokurov , and Calum Spicer . Positivity of the Moduli Part . arXiv e-prints , page arXiv:2111.00423, October 2021

arXiv 2021

[2] [2]

Boundedness and K^2 for log surfaces

Valery Alexeev. Boundedness and K^2 for log surfaces . International Journal of Mathematics , 05(06):779--810, 1994

work page 1994

[3] [3]

The moduli b-divisor of an lc-trivial fibration

Florin Ambro. The moduli b-divisor of an lc-trivial fibration . Compositio Mathematica , 141(2):385–403, 2005

work page 2005

[4] [4]

Singular toric Fano varieties

Alexandr Borisov and Lev Borisov. Singular toric Fano varieties . Sbornik Mathematics , 75:277--283, 1993

work page 1993

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