REVIEW 3 major objections 4 minor 12 references
Pathological MMP singularities as $\alpha_p$-quotients
T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read In every positive characteristic p, quotients of affine space by an alpha_p-action produce canonical or terminal singularities that fail S3, and stable families with non-S2 special fibers that block proper KSBA-type moduli when p ≤ n.
desk verdict Genuinely new non-S3 terminal singularities and stable families via alpha_p-quotients, with explicit computations that check out; the one serious gap is an unproved Q-factoriality assertion that the discrepancy classification depends on. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the rank-one 1-foliation F on affine space generated by the p-closed quadratic derivation ∂ = Σ_{i=1}^n $x_i^{2}$ ∂_{x_i}, which satisfies ∂^[p] = 0 and so defines an alpha_p-action; the quotient A^n/F is the singularity under study. Its singularities are computed through the discrepancy formula for quotients by 1-foliations (Theorem 2.2.8): for a divisor E over the source with image F over the quotient, a(F; Y, Δ_Y) equals a(E; X, Δ) + (p − 1)a(E; F) when E is foliation-invariant and 1/p times that otherwise. After a single blow-up of the origin, the pulled-back foliation becomes log canonical, so the formula reduces the discrepancy computation to discrepancies of a log smooth pair (X, (p − n)E); the inequality n ≥ p (resp. n ≥ p+1) forces the singularity to be canonical (resp. terminal). For the families, the analogous relative foliation ∂_m = Σ $x_i^{2}$ ∂_{x_i} + t^m μ(y) ∂_y is blown up repeatedly until it is log canonical, yielding crepant equations whose log canonical threshold is n + 1 ≥ p. The identification of fibers of the quotient family with quotients of fibers (Theorem 2.2.10) is what certifies non-S2ness at t = 0.
What would settle it
Compute $H^{2}$_m(O_{Y,0}) for Y = $A^{{p+1}}$/∂ with ∂ = Σ_{i=1}^{p+1} $x_i^{2}$ ∂_{x_i} in characteristic p: the paper predicts a nonzero module with nilpotent Frobenius, so depth exactly 2 and not F-injective. If this local cohomology instead vanishes, the non-S3 claim collapses; likewise, checking the claimed equality a(E; F) = −1 for the exceptional divisor of the first blow-up of the quadratic foliation is a direct calculation on which every subsequent discrepancy inequality depends.
Extended reading notes
Core claim
The paper establishes that alpha_p-quotients, studied through rank-one foliations, are a uniform source of pathological MMP singularities in every positive characteristic. Concretely, Theorem 3.0.2 states that for each p > 0 there is an isolated Q-factorial singularity (0 ∈ Y) that is canonical of dimension max{p,3} and terminal of dimension p+1 and is not S3; Theorem 4.1.3 produces locally stable families Y → $A^{1}$ of relative dimension max{p,3} with Y_t smooth for t ≠ 0 and Y_0 reduced but non-S2; Theorem 4.1.7 compactifies them to projective stable families of pairs (Y, (1/p)H) → $A^{1}$ with mu_p-quotient general fibers and reduced non-S2 central fiber, stable under finite flat base change. The moduli consequence, Theorem A.0.12, is that in characteristics p ≤ n there is no proper potential KSBA, KSBA-CM, or KSBA-F-injective moduli stack of n-dimensional stable pairs with the usual boundary coefficients; the volumes of the pathological general fibers form a dense subset of (0, ∞). The method also yields a locally stable 3-fold family in characteristic 3 from the derivation $y^{3}$ ∂_x + x ∂_y + t ∂_z.
Load-bearing premise
All of the discrepancy, stability, and moduli conclusions reduce through the quoted formula for discrepancies of quotients by rank-one foliations and the criterion that foliations with multiplicative singularities are log canonical; if either is false, the constructed quotients are not shown to be terminal, canonical, or locally stable.
Editorial extensions
If this is right
- For every p > 0 there is an isolated terminal non-S3 singularity of dimension p+1, improving the previously known asymptotic dimension bound from 2p+2 to p+1 and giving a threefold example in characteristic 2.
- For every p > 0 there is an isolated canonical non-S3 singularity of dimension max{p,3}; in particular canonical singularities need not be Cohen-Macaulay in any positive characteristic.
- Locally stable one-parameter families of relative dimension max{p,3} can have smooth general fibers and a reduced non-S2 central fiber, so the characteristic-0 statement that fibers of locally stable families are S2 fails in every positive characteristic.
- Projective stable families of pairs (Y, (1/p)H) → A^1 exist whose general fibers are mu_p-quotient (hence klt, Cohen-Macaulay, and F-injective) and whose central fiber is reduced but non-S2; they remain normal and stable after any finite flat base change C → A^1.
- In characteristics p ≤ n, the pathological families obstruct the valuative criterion of properness for every potential KSBA, KSBA-CM, and KSBA-F-injective stack of n-dimensional stable pairs, with a dense set of volumes.
Reading between the lines
- The dimensional bound max{p,3} is tied to the quadratic derivation; other p-closed derivations, such as y^3 ∂_x + x ∂_y used in Section 4.2, may produce similar pathologies in lower fixed dimension for larger p, though the required blow-up sequences grow.
- The construction shows that discrepancy-based singularity classes (terminal, canonical, lc) are largely independent of Serre depth in positive characteristic: the same quotient can be terminal and fail S3, so MMP singularity classes alone do not control cohomological depth.
- For moduli theory, the failure is robust rather than isolated: because volumes are dense and the family is stable under finite flat base change, the obstruction to properness cannot be removed by imposing CM or F-injectivity on general fibers; any positive-characteristic KSBA theory will need a different condition on the total family or special fiber.
- A natural testable extension, raised in the paper's Question 1.1.1(a), is whether p ≫ n restores properness; the discrepancy formulas' dependence on p − n suggests high characteristics should behave closer to characteristic 0, but the paper leaves this open.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper constructs, in every positive characteristic p, α_p-quotient singularities with pathological MMP behavior. The main results are: (Theorem 3.0.2) isolated Q-factorial canonical (dimension max{p,3}) and terminal (dimension p+1) non-S3 singularities; (Theorems 4.1.3, 4.1.7, 4.2.3) locally stable and stable families of relative dimension at least max{p,3} whose special fiber is reduced but non-S2 while general fibers are smooth or have only µ_p-quotient singularities; and (Theorem A.0.12) a consequence ruling out properness of natural KSBA-type moduli stacks for Cohen–Macaulay or F-injective pairs when p ≤ n. The technical engine is discrepancy computation for quotients by rank-1 foliations, imported from the author's [Pos23], applied to explicit p-closed derivations such as ∂ = Σ x_i^2 ∂_{x_i}.
Significance. If the missing justifications identified below are supplied, these are substantial results. They improve the known dimensional lower bounds for terminal non-CM singularities in positive characteristic, provide new examples of locally stable families with non-S2 fibers that differ from Kollár's, and give a clean argument against naive KSBA compactifications in small characteristic. The constructions are fully explicit and the discrepancy computations are checkable line by line; the paper does not rely on numerical fitting or ad hoc parameters. The main obstruction to accepting the results as they stand is the unproved Q-factoriality assertion and the reliance on an extension of the discrepancy formula for sub-pairs that is asserted but not demonstrated.
major comments (3)
- [§3, Theorem 3.0.2(a)] The proof begins "By construction Y is normal and Q-factorial" and cites [Pos23, Lemma 2.5.10] only for regularity away from the origin. Q-factoriality is load-bearing: the statements in Theorem 3.0.2(c), Proposition 3.0.4, and the crepant equations in Sections 4.1 and 4.2 all presuppose that K_Y, or K_Y + Δ, is Q-Cartier, per Definition 2.1(d). For an α_p-quotient, the invariant ring is not a direct summand of the regular ring, so the characteristic-0 descent argument for divisor classes does not apply. Please provide a proof of Q-factoriality, or a precise reference, or modify the statements to only assert Q-Cartierness of K_Y if that is all the discrepancy computations require.
- [§2.2, Theorem 2.2.8] The theorem is stated for a normal sub-pair (X,∆), but the proof says only that [Pos23, Theorem 4.2.5] proved the effective case and that effectiveness is not needed. This extension is not automatic and is essential: later computations use non-effective boundaries, e.g., (p−n)F with p−n < 0 in Theorem 3.0.2, and the negative combinations −Σ a_i F_i in Theorems 4.1.3 and 4.1.7. Please provide the full sub-pair statement with a proof, or at least a detailed account of which steps in [Pos23, Theorem 4.2.5] extend verbatim.
- [§4.1, Theorem 4.1.7, Step 3] The proof reduces local stability after an arbitrary finite flat base change C → A^1 to the case of iterated Frobenius base changes, citing [Kol23b, 2.15.5] and [HZ20]. The reduction is not spelled out, and the parenthetical alternative in the proof is only a sketch. Since the "Moreover" clause of Theorem 4.1.7 is part of the theorem statement, please either state the precise lemma from the cited references that applies, or carry out the DVR base-change argument with the bookkeeping for the Q-boundary (1/p)H.
minor comments (4)
- [§4.1, Theorem 4.1.3] The sentence "Since Y is S2, the Cartier divisor Y0 is S1" uses the S2 property of Y, but normality or S2 of Y is not explicitly established before this point; please cite the relevant criterion (e.g., [Pos23, Remark 2.5.4]) or prove it.
- [§4.1, Theorem 4.1.7, Step 2] The text says "We let H ⊂ Y be the prime divisor with support q(H)" although H is a Q-divisor; this should be formulated as the pushforward (or image) of a Q-divisor, not a prime divisor.
- [§4.2, Claim 4.2.1] The formula in Claim 4.2.1 concludes with an exponent (5p−3)/2; for clarity, please note explicitly that this is an integer for odd p and state the characteristic range (p > 2) before the claim.
- [Appendix A, Definition A.0.6] In condition (c), the values of a stack on a regular curve are described as a set, but a stack should assign a groupoid; if only isomorphism classes are intended, this should be stated.
Circularity Check
No significant circularity: the quotient constructions and discrepancy computations are explicit and self-contained, with load-bearing but independent imports from the author's prior work.
full rationale
The derivation chain is explicit and not circular. The paper constructs p-closed derivations (Claims 3.0.1, 4.1.1), takes their quotients, and computes the discrepancies of the quotients by reducing to discrepancies on blow-ups via Theorem 2.2.8. That theorem is quoted from the author's earlier work [Pos23, Theorem 4.2.5], as are the companion foliation tools (Propositions 2.2.4 and 2.2.7, Theorem 2.2.10). These are parameter-free general statements whose assumptions do not include the existence or dimension of the pathological singularities being proved; they are imported computational lemmas, not the paper's conclusions. No fitted constant or post-hoc normalization is renamed as a prediction: the constructions are closed-form quotients by explicit derivations, and the claimed failure of properness for KSBA moduli stacks (Theorem A.0.12) follows by feeding these families into the definition of a potential moduli stack, not by assuming that failure. The only notable issue is a non-circular correctness gap: the proof of Theorem 3.0.2 states 'By construction Y is normal and Q-factorial' without giving an argument, although Q-factoriality (or at least Q-Cartierness of K_Y) is needed for the discrepancy computations; this is a possible missing proof, not a circular reduction, and it does not affect the circularity score.
Assumptions & free parameters
assumptions (4)
- domain assumption Theorem 2.2.8: discrepancy formula for 1-foliation quotients (from [Pos23, Theorem 4.2.5]).
- domain assumption Proposition 2.2.4: a rank-1 foliation on a regular variety is lc iff it has only multiplicative singularities (from [Pos23, Corollary 1]).
- domain assumption Theorem 2.2.10: for a family of 1-foliations, the natural map X_b/F|_{X_b} to (X/F)_b is an isomorphism iff (X/F)_b is S2 (from [Pos23, Proposition 5.2.4, Corollary 5.2.5]).
- standard math Standard tools: Zariski's discrepancy lemma (Lemma 2.1.2), toroidal resolution [KKMSD73], and birational invariance of higher direct images [CR11, Theorem 1].
Cite this review
Pith. "Pith review of Pathological MMP singularities as $\alpha_p$-quotients." pith.science (2026). https://pith.science/paper/S4MZ652M
@misc{pith2026250101179,
author = {Pith},
title = {Pith review of: Pathological MMP singularities as $\alpha_p$-quotients},
year = {2026},
howpublished = {\url{https://pith.science/paper/S4MZ652M}},
note = {Machine review of arXiv:2501.01179}
}
abstract
We construct pathological examples of MMP singularities in every positive characteristic using quotients by $\alpha_p$-actions. In particular, we obtain non-$S_3$ terminal singularities, as well as locally stable (respectively stable) families whose general fibers are smooth (respectively klt, Cohen--Macaulay and $F$-injective) and whose special fibers are non-$S_2$. The dimensions of these examples are bounded below by a linear function of the characteristic.
Reference graph
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Reviewed August 10, 2026 · model on record in the stance chip above.
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