{"id":"db77ee79-8e7a-4774-92ab-26d404867e0c","arxiv_id":"2501.01179","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every positive characteristic, the author constructs non-S3 terminal singularities of dimension p+1 and stable families with klt, Cohen-Macaulay, F-injective general fibers but non-S2 special fibers.","lead":"This paper uses quotients by tiny group actions to build new 'pathological' singularities in positive characteristic geometry, where minimal model program singularities can fail expected properties like Cohen-Macaulayness. It also constructs stable families whose limit fiber is not depth-two, blocking the natural compactification of moduli spaces of stable pairs.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unproved Q-factoriality assertion in Theorem 3.0.2(a) is load-bearing: the discrepancy classifications are only defined if K_Y is Q-Cartier, and this is not automatic for alpha_p-quotients.","rationale":"The reader's weakest_assumption focuses on the imported discrepancy formula Theorem 2.2.8 from [Pos23]. I agree that this formula is a major external dependency, and the paper would be easier to evaluate if the proof were reproduced or a precise statement with all hypotheses were included. However, the single most load-bearing gap in the text itself is the unsupported claim in the proof of Theorem 3.0.2 that Y = A^n/F is Q-factorial 'by construction'. This is not a minor strengthening: the entire MMP classification in the paper is expressed through discrepancies of K_Y, and the conventions in Definition 2.1(d) require K_Y + Delta to be Q-Cartier before any discrepancy can be computed. For alpha_p-quotients the quotient is finite and radical but not a direct summand inclusion of invariant rings, so the standard descent argument that gives Q-factoriality for finite group quotients in characteristic zero does not apply. If the class group is nonzero and non-torsion, the stated theorem and the stability statements in Section 4 would need reformulation. My spot checks of the weighted blow-up computation in Proposition 3.0.4 and the crepant bookkeeping in Theorem 3.0.2 are internally consistent, so I do not see a contradiction inside the main argument; the issue is a missing foundational proof. The concrete test above would settle whether the assertion is true in the first nontrivial case, and thus whether the concern actually lands.","tokens_in":29603,"tokens_out":15166,"duration_ms":163442,"concrete_test":"Take p = 3, n = 3, and compute a presentation of the invariant ring R = F_3[x,y,z]^{x^2 d_x + y^2 d_y + z^2 d_z} using, for example, Singular or Macaulay2. Then compute the divisor class of the image of the hyperplane (x = 0) in Spec R, or directly test whether the canonical module omega_R becomes locally free after a finite twist. If that divisor class has infinite order, or if no positive multiple is Cartier, Theorem 3.0.2(a) fails. If the computation shows Q-factoriality for this case, run the same check for the smallest remaining cases n = p and n = p + 1 in the relevant characteristic to confirm the assertion used in the proof.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 3.0.2 begins 'By construction Y is normal and Q-factorial' and gives no argument or citation for Q-factoriality; [Pos23, Lemma 2.5.10] is cited only for regularity away from the origin. This matters because every discrepancy statement in the paper (Theorem 3.0.2(c), Proposition 3.0.4, and the lc conclusions in Sections 4.1 and 4.2) is formulated with respect to a Q-Cartier canonical divisor, by the conventions of Definition 2.1(d), and Theorem 2.2.8 computes discrepancies only after one can write K_Y, or K_Y + Delta, as a Q-Cartier class. Q-factoriality is not automatic for alpha_p-quotients: the quotient map is a radical finite morphism and the invariant ring is not a direct summand, so the characteristic-0 descent argument for divisor classes does not apply. If some Weil divisor, for example the image of a coordinate hyperplane, has no positive multiple that is Cartier, then K_Y is not Q-Cartier and the canonical/terminal classification of the constructed singularity is not defined by the paper's own definitions. The same issue propagates to the locally stable families of Section 4, whose crepant relations presuppose that the relevant log canonical divisors are Q-Cartier.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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}.","tokens_in":29739,"tokens_out":10968,"duration_ms":102835,"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":[{"comment":"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.","section":"§3, Theorem 3.0.2(a)"},{"comment":"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.","section":"§2.2, Theorem 2.2.8"},{"comment":"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.","section":"§4.1, Theorem 4.1.7, Step 3"}],"minor_comments":[{"comment":"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.","section":"§4.1, Theorem 4.1.3"},{"comment":"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.","section":"§4.1, Theorem 4.1.7, Step 2"},{"comment":"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.","section":"§4.2, Claim 4.2.1"},{"comment":"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.","section":"Appendix A, Definition A.0.6"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically interesting and the constructions are explicit, but the missing proof of Q-factoriality in Theorem 3.0.2 is a genuine load-bearing gap, and the sub-pair extension of Theorem 2.2.8 is asserted with an insufficient justification. Both are likely fixable, so I recommend major revision rather than rejection. The reduction in Theorem 4.1.7 Step 3 also needs a precise reference or a complete argument. If the author can supply these, the paper would be a strong contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuinely original paper that does what it says—constructs terminal non-S3 singularities in dimension p+1 for every positive characteristic, improving Totaro's 2p+2 bound, and stable families with klt, Cohen–Macaulay, F-injective general fibers and non-S2 special fiber. The quadratic-derivation quotient method is simple and the discrepancy computations are explicit; I spot-checked the weighted blow-up in Prop. 3.0.4 and the non-S2 argument in Thm. 4.1.3, and they work. The paper deserves a serious referee.\n\nThe soft spot is exactly where the reader put it. Theorem 3.0.2 opens with \"By construction Y is normal and Q-factorial\", with no proof or citation. For alpha_p-quotients the invariant ring is not a direct summand, so the usual characteristic-0 descent of divisor classes fails; Q-factoriality is not automatic. The discrepancy formulas need at least K_Y Q-Cartier, and the crepant relation K_Z + (p-n)F = b^*K_Y is written as an equality of Q-Cartier divisors. If K_Y were not Q-Cartier, the canonical/terminal classification would be undefined by the paper's own conventions. I don't think the claim is false—the quadratic derivation is homogeneous and the quotient may well be Q-factorial (or at least K_Y Q-Cartier)—but a proof has to be supplied. This is the one load-bearing gap I found.\n\nSecond, the paper leans heavily on the author's own [Pos23] for Theorem 2.2.8 and Proposition 2.2.4. That is self-citation, but the cited results are exactly what make the whole construction work. [Pos23] is listed as \"to appear in Nagoya Math. J.\", so it is not a black hole, but the referee will need to verify those statements. Smaller items: Prop. 3.0.5 relies on [BBK24] and [Bau23], also preprints, and the KSBA appendix is openly tentative about the moduli definition, though the density argument in Thm. A.0.12 is fine.\n\nOverall, the central mathematical construction is coherent, the new examples are real, and the dimension improvement is meaningful. The Q-factoriality gap should be fixed, and the imports from [Pos23] should be stated explicitly as such. This is publishable after revision; I'd send it to a strong referee.","headline":"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.","tokens_in":790,"tokens_out":940,"would_cite":true,"duration_ms":32957,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14B05","14E30","14J17"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["positive characteristic","MMP singularities","terminal singularities","Serre condition S3","alpha_p-actions","1-foliations","stable families","KSBA moduli"],"falsifier":"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.","tokens_in":29251,"feed_emoji":"🧮","tokens_out":12147,"duration_ms":102356,"temperature":0.7,"pith_summary":"In every positive characteristic p, the paper constructs isolated quotient singularities of affine space that are terminal in dimension p+1 and canonical in dimension max{p,3} yet fail Serre's condition S3. The quotients are taken by the alpha_p-action generated by the quadratic derivation ∂ = Σ $x_i^{2}$ ∂_{x_i}. The same construction, run in families, gives locally stable families with smooth general fibers and reduced non-S2 central fibers, and projective stable families of pairs whose general fibers have only mu_p-quotient singularities while the special fiber is reduced but non-S2. Since such a central fiber cannot be replaced by an S2 (in particular Cohen-Macaulay or F-injective) fiber after any finite base change, the paper concludes that no proper KSBA-type moduli stack of n-dimensional Cohen-Macaulay or F-injective stable pairs exists when p ≤ n. An additional construction produces a locally stable family of threefolds in characteristic 3.","feed_headline":"Terminal singularities that are not S3 in every positive characteristic","feed_subtitle":"The same quotient construction yields stable families with non-S2 special fibers, blocking proper KSBA moduli for p ≤ n.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the discrepancy formula for quotients by rank-one foliations (Theorem 2.2.8), the lc criterion for multiplicative singularities (Proposition 2.2.4), and the fiber-quotient identification (Theorem 2.2.10).","marker":"[Pos23]"},{"why":"Provides the standard discrepancy and log canonical framework, Corollary 2.11 for log smooth thresholds, and the definitions of locally stable and stable families used throughout.","marker":"[Kol13]"},{"why":"Gives the Zariski lemma (Lemma 2.45) used in Lemma 2.1.2 to bound discrepancies by codimension, a step in proving terminality and canonicity.","marker":"[KM98]"},{"why":"Supplies the definitions and base-change properties of stable families of pairs used in Theorem 4.1.7 and in the KSBA moduli appendix.","marker":"[Kol23b]"},{"why":"Provides the blow-up and weighted-blow-up techniques for simplifying foliation singularities invoked in Remark 4.1.9 and the p = 2 variant of Theorem 4.1.7.","marker":"[Pos24b]"},{"why":"Gives the fact that on a supersingular elliptic curve a global vector field ω satisfies ω^[p] = 0, used to build the compactified families in Lemma 4.1.8.","marker":"[KM85]"}],"fun_headline_variants":["α_p-quotients yield non-S3 terminal singularities in all characteristics","No proper moduli for stable pairs in char p ≤ n","Stable families with non-S2 fibers from α_p-quotients","Blocking KSBA moduli via α_p-quotients in char p ≤ n"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["α_p-quotients yield non-S3 terminal singularities in all characteristics","No proper moduli for stable pairs in char p ≤ n","Stable families with non-S2 fibers from α_p-quotients","Blocking KSBA moduli via α_p-quotients in char p ≤ n"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001438,"raw_usage":{"total_tokens":5771,"prompt_tokens":897,"completion_tokens":4874,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":513,"completion_tokens_details":{"reasoning_tokens":4793}},"tokens_in":513,"tokens_out":4874,"duration_ms":32098,"temperature":1.0,"reasoning_tokens":4793,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:35:07.172450+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}