{"id":"31f7f187-d1cb-4b43-ae1f-ecf50af98227","arxiv_id":"2506.01007","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Projective globally F-split semi-log canonical K-trivial surfaces in characteristic p > 2 admit strong equisingular liftings over the Witt vectors.","lead":"This paper proves that every projective semi-log canonical, globally F-split surface with trivial canonical class over a field of characteristic p > 2 can be lifted, in an equisingular way, to a surface over the ring of Witt vectors, whose fraction field has characteristic 0. The result extends a prior lifting theorem for normal surfaces to singular, non-normal surfaces, a step toward understanding the boundary of moduli spaces of Calabi-Yau surfaces in mixed characteristic.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 7.7's equivariant minimal model list rests on a characteristic-zero classification; the paper does not justify it for p>2, and the 4A1 lifting argument depends on it.","rationale":"The paper is a serious and substantial extension of [BBKW24], and the internal structure of the proof is mostly transparent: the reduction to the normalization pair, the handling of index 1, the reduction to the 4A1 case, and the long case analysis in Section 7 are all carefully developed. The reader's MODERATE/ACCEPT verdict is reasonable if Lemma 7.7 is sound in positive characteristic. My stress-test focuses on the single external input that the 4A1 case depends on: the classification of μ2-equivariant minimal models of the rational index-one cover. Lemma 7.7 cites [BB00, Theorem 1.4] without discussing characteristic; if that classification is only established in characteristic zero, the transfer to p>2 is not automatic in general, even for tame group actions, because the list of minimal rational G-surfaces can depend on the characteristic (e.g., del Pezzo cases and conic bundle structures). The paper does provide a direct proof of Lemma 7.5, so the connectedness principle is less of a concern; the missing justification is specifically the enumerative list in Lemma 7.7. I am not asserting that a counterexample exists, only that the proof of the main theorem has a load-bearing unverified input. The proposed check—comparing Lemma 7.7 with a positive-characteristic classification and attempting to realize a potential missing del Pezzo degree-4 case—would settle the matter. If the check confirms the list, ACCEPT is appropriate; until then, CONDITIONAL is the honest verdict.","tokens_in":65305,"tokens_out":13901,"duration_ms":151244,"concrete_test":"Independently verify Lemma 7.7 over algebraically closed fields of characteristic p>2. Concretely: (1) Search the positive-characteristic equivariant minimal model literature (e.g., Iskovskikh–Manin and Prokhorov [Pro21]) for a classification of minimal rational μ2-surfaces with Pic^G of rank 1 and compare it with the list in Lemma 7.7. (2) In particular, try to construct over an algebraically closed field of characteristic 3 or 5 a smooth del Pezzo surface U of degree 4 with an involution σ that preserves an ordinary elliptic anticanonical divisor E_U, has no σ-invariant exceptional curves, and has rational quotient U/σ. If such U exists, form S = U/σ and check whether S is a globally F-split slc CY surface whose conductor contains a 4A1-curve; its existence would falsify Lemma 7.7. If no such example exists and the list is confirmed in p>2, the concern is resolved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 7 hinges on Lemma 7.7, which asserts that when the index-one cover T is rational, the μ2-equivariant minimal model U of the log resolution (T', E'+Γ') is one of five types: P2; del Pezzo of degree 2 with the Geiser involution; degree 1 with the Bertini involution; P1×P1 with the switch; or a μ2-conic bundle. The only justification given is: 'Since U is rational and minimal, the action U ⟲ μ2 is one of those listed in [BB00, Theorem 1.4].' [BB00] is a classification of birational involutions of P2 over an algebraically closed field, in the literature stated in characteristic zero; the paper works over algebraically closed k of characteristic p>2 and cites no positive-characteristic analogue for this enumerative step. The MMP for surfaces in positive characteristic ([Tan14,18]) does not by itself yield the list, and [Pro21] is cited for general equivariant MMP facts rather than for this classification. A missing case—for instance, a del Pezzo surface of degree 4 carrying an order-2 automorphism with no invariant (-1)-curves and an invariant ordinary anticanonical elliptic curve—would fit the hypotheses of Proposition 7.4 but is not covered by Sections 7.6-7.11. Since Proposition 7.12 and Proposition 7.14 reduce the 4A1 lifting problem to exactly these equivariant models, any such missing case would block the lifting of the gluing involution τ and the proof of Theorem 1.1. This is a concrete correctness risk about the completeness of the classification, not a disagreement with consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that every projective semi-log canonical globally F-split surface X over an algebraically closed field of characteristic p>2 with K_X numerically trivial admits a strong semi-log canonical (equisingular) lifting over the Witt vectors W(k). The proof reduces the non-normal case to the normal case of [BBKW24] by showing, in Proposition 4.9, that a strong slc lift is equivalent to a strong log lift of the normalization pair together with a lift of the gluing involution. The main technical work is the 4A1 case: the authors introduce a canonical lifting of the log pair (P^1, 1/2 sum of four points) via an ordinary elliptic double cover, analyze the index-one cover of the normalization pair, classify its equivariant minimal models, and construct equivariant liftings for each model type in Sections 7.6-7.11. A separate argument in Section 7.11 excludes the index-four case, completing the proof of Theorem 6.3 and hence of Theorem 1.1.","tokens_in":65577,"tokens_out":15789,"duration_ms":169400,"significance":"If correct, the theorem confirms, for slc K-trivial surfaces, the expectation that globally F-split varieties admit equisingular deformations to characteristic zero, extending the normal surface result of [BBKW24] to a non-normal setting that is central for moduli theory of K-trivial surfaces in mixed characteristic. The paper is structurally careful: the main theorem is reduced to a small number of explicit geometric classifications, and the authors give constructive lifting arguments for each case, using canonical lifts of ordinary elliptic curves as the key functorial input. The proof is essentially self-contained modulo the cited normal-surface and canonical-lift results, and I found no circularity: the inputs [BBKW24], [MS87], and the MMP references are independent prior results. The main uncertainty is the completeness of the equivariant minimal-model classification in characteristic p>2, discussed below.","major_comments":[{"comment":"The five-case classification of the μ2-equivariant minimal model U is justified only by the sentence 'Since U is rational and minimal, the action U ⟲ μ2 is one of those listed in [BB00, Theorem 1.4].' The cited theorem of Bayle–Beauville is, as usually stated, a classification of birational involutions of the complex projective plane, whereas the paper works over an algebraically closed field k of characteristic p>2. No positive-characteristic analogue is given, and [Pro21] is cited for general equivariant MMP facts rather than for this enumerative classification. This issue is load-bearing: Propositions 7.12 and 7.14 reduce the 4A1 lifting problem exactly to the models listed in Lemma 7.7, and Sections 7.6-7.11 cover precisely that list and no others. The authors should either cite a classification that is valid over algebraically closed fields of characteristic p>2, or add a short proof (for instance, by observing that a smooth G-minimal del Pezzo surface with ρ^G=1 and an order-two automorphism can only have degree 1 or 2, since orbit sums of lines give invariant classes independent of K, and otherwise the surface is a conic bundle).","section":"§7.4, Lemma 7.7"}],"minor_comments":[{"comment":"The abstract in the submission metadata states 'characteristic p>0', while Theorem 1.1 and the body of the paper require p>2; please make these statements consistent.","section":"Abstract"},{"comment":"The section title contains the typo 'lifings'; it should read 'liftings'.","section":"§7"},{"comment":"The footnote says that a point on a component of Γ_Ui can be lifted because 'they are rational', but Lemma 7.9 allows Γ_U to be a regular curve of genus one. The lifting argument is still valid by Hensel's lemma since the relevant divisors are smooth over W(k), so the justification should be corrected rather than the conclusion changed.","section":"§7.5, Proposition 7.14, footnote 9"}],"recommendation":"major_revision","confidential_remarks":"The paper is a strong contribution and the central argument appears coherent. My main concern is the characteristic-zero citation in Lemma 7.7; once the authors supply a valid reference or a short proof for the p>2 equivariant classification, I would be willing to accept. I do not find the skeptical scenario of a missing del Pezzo degree 4 case convincing: such a surface would not be μ2-minimal because orbits of lines give invariant classes independent of K, so it would not survive the equivariant MMP; the real issue is the missing justification for the classification in the stated generality."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main theorem: this is a genuine advance. It extends the equisingular lifting result of Bernasconi–Brivio–Kawakami–Witaszek from normal surfaces to demi-normal slc K-trivial surfaces in characteristic p > 2, and it does so by a structurally coherent argument. The reduction of the main theorem to lifting the gluing involution (Proposition 4.9) is clean. The canonical lift of the log Calabi–Yau pair (P1, 1/2 sum of four points) via an ordinary elliptic curve is new, and the µ2-equivariant case analysis in Section 7 is explicit and largely convincing. Credit is due for the index-one-cover analysis and for the careful handling of the 4A1 cases.\n\nThe soft spot is real and concentrated in Lemma 7.7. The lemma is load-bearing for the 4A1 argument, and its proof is essentially one sentence: the µ2-minimal model U is rational and minimal, so the action is one of those listed in [BB00, Theorem 1.4]. That is a characteristic-zero classification of birational involutions of P2, and the paper never explains why the same list is complete over an algebraically closed field of characteristic p > 2. The equivariant MMP for surfaces in positive characteristic, cited elsewhere in the paper, does not by itself yield that enumerative statement. The stress-test worry about a missing del Pezzo case is probably not a real counterexample — if such an example existed, it would contradict the characteristic-zero list already, and the list is accepted there — but the missing justification is a genuine gap. A referee should ask the authors to either prove the classification for tame order-2 actions in characteristic p > 2 or supply a positive-characteristic reference. This is fixable, but it is not cosmetic.\n\nOther concerns are minor. The reliance on [BBKW24] for the normal case and [MS87] for canonical lifts is appropriate; there is no circularity. The paper explicitly leaves p = 2 open, which is honest. The writing is dense and the paper is long, but the structure helps. The main theorem is significant and the proof is credible overall.\n\nVerdict: send it to peer review. The referee's main job should be to pin down Lemma 7.7 and the positive-characteristic classification. If that patch holds, the paper is publishable.","headline":"Real extension of the normal-surface result to the non-normal slc boundary; the proof is credible, but the char-p transfer of the involution classification in Lemma 7.7 is under-justified.","tokens_in":66172,"tokens_out":5564,"would_cite":true,"duration_ms":63951,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14G17","14J32"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every globally F-split semi-log canonical K-trivial surface in characteristic p>2 lifts equisingularly over the Witt vectors.","keywords":["Calabi–Yau surfaces","global F-splitting","lifting to characteristic zero","positive characteristic","mixed characteristic","semi-log canonical surfaces","equisingular deformation","Witt vectors"],"falsifier":"Look for a projective globally F-split slc surface X over an algebraically closed field of characteristic p>2 with K_X ≡ 0 whose normalization pair has a rational boundary component with four A1-singularities, and compute the μ2-equivariant minimal model of its index-one cover: if that model is not one of the five types in Lemma 7.7, or if the four branch points on the canonical lift of (\\mathbb{P}^1, \\frac{1}{2}\\sum q_i) carry no involution lifting τ, then Theorem 1.1 is false.","tokens_in":65042,"feed_emoji":"🧩","tokens_out":9266,"duration_ms":93440,"temperature":0.7,"pith_summary":"The paper proves a lifting theorem: in characteristic p>2, any projective surface with the mildest singularities of moduli-theoretic interest — semi-log canonical (slc) — whose canonical class is numerically trivial and which is globally F-split (a Frobenius-positivity condition) admits an equisingular deformation to characteristic 0 over the ring W(k) of Witt vectors. This extends the known theorem for normal globally F-split surfaces to non-normal slc surfaces, the kind that appear as boundary points in moduli spaces of K-trivial surfaces. The proof shows that the only genuine difficulty in the non-normal case is lifting the gluing involution that identifies the two sheets of the normalization along the conductor, and it resolves this by constructing a strong log lift of the normalization on which the involution lifts.","feed_headline":"F-split K-trivial surfaces lift to characteristic 0","feed_subtitle":"Every such singular surface in characteristic p>2 admits an equisingular deformation over the Witt vectors.","key_machinery":"The central object is the normalization triple (X^\\nu, D, τ): X^\\nu is the normalization of the demi-normal surface X, D is the conductor divisor (the double locus), and τ is the induced involution on the normalization D^\\nu of D, a log involution of (D^\\nu, Diff_{D^\\nu}(0)). The proof's criterion (Proposition 4.9) says that a strong slc lifting of X is exactly a strong log lifting of (X^\\nu, D) with K_{X^\\nu}+D Q-Cartier together with a lift of τ. To lift τ, the paper builds canonical liftings of the two boundary types that matter: ordinary genus-one curves, using the classical canonical lift with Frobenius, and the four-marked rational pair (\\mathbb{P}^1, \\frac{1}{2}\\sum_{i=1}^4 q_i), whose canonical lift is obtained from a degree-two cover by an ordinary elliptic curve. In the hardest case, where D has a 4A1-curve (a rational component along which S has four A1-singularities), the proof passes to the index-one cover, classifies its μ2-equivariant minimal models — del Pezzo surfaces of degree 1 or 2, the projective plane, the product of two projective lines, or conic bundles — and lifts each model equivariantly over W(k).","core_discovery":"On the paper's own terms, the discovery is Theorem 1.1: if k is algebraically closed of characteristic p>2 and X is a projective semi-log canonical, globally F-split surface with K_X ≡ 0, then there exists a proper strong semi-log canonical lifting X of X over W(k). A strong slc lifting is an equisingular deformation in the precise sense of Section 4: the special fiber is X, the total space is demi-normal, and its normalization is a strong log lifting of the normalization pair of X, with K_{X^\\nu}+D Q-Cartier. Equivalently, the paper shows that the normalization pair can be lifted together with the order-two gluing involution τ on the conductor double cover, so that the quotient of the lifted pair reconstructs a lift of X.","pith_inferences":["The construction singles out the four-marked rational boundary as the only genuine obstruction: a natural next test is to classify F-split slc K-trivial surfaces whose boundary is exactly one 4A1-curve and to check whether the canonical lift produced here is forced or admits alternatives.","The canonical lift of (\\mathbb{P}^1, \\frac{1}{2}\\sum q_i) via an ordinary elliptic double cover is a transferable tool; it should apply to other log Calabi–Yau pairs and to Enriques-type quotients in positive characteristic, where the liftability problem has the same flavor.","The theorem is evidence for the broader expectation that globally F-split varieties lift over W(k); the strategy suggests that in higher dimensions the analogous obstruction would be the equivariant classification of index-one covers, not Frobenius-positivity itself."],"forward_implications":["Every globally F-split slc K-trivial surface in characteristic p>2 becomes the special fiber of a proper, locally stable family over W(k); the singularities, including the four A1 points on each 4A1-curve, deform equisingularly rather than smoothing.","In the 4A1 case the proof forces the Cartier index of K_{X^\\nu}+D to be exactly 2; index-4 covers cannot occur, so only μ2-equivariant models are needed.","If the normalization boundary has no four-marked rational component, lifting the gluing involution is automatic, so the non-normal theorem follows directly from the normal case.","For moduli theory, the F-split slc K-trivial surfaces in characteristic p>2 are specializations of the characteristic-0 slc boundary, assuming the relevant moduli space is proper.","The theorem stops at p>2 and K-triviality; p=2 and non-CY cases are explicitly left open, with the paper noting that in p=2 inseparable nodes are already excluded by F-splitting."],"supporting_citations":[{"why":"Supplies the strong log lifting theorem for normal globally F-split surfaces, together with the lifting methods that the non-normal proof generalizes.","marker":"[BBKW24]"},{"why":"Provides canonical liftings of ordinary elliptic curves and the Frobenius-compatible lifting framework used to lift involutions on genus-one boundary components.","marker":"[MS87, Appendix]"},{"why":"Gives the classification of birational involutions on the projective plane that underlies the classification of μ2-equivariant minimal models in Lemma 7.7.","marker":"[BB00]"},{"why":"Supplies the gluing and demi-normality results in mixed characteristic used to convert a lifted normalization with a lifted involution into a strong slc lifting.","marker":"[Pos25]"},{"why":"Provides the gluing theory for slc surfaces in positive characteristic, including the normalization triple and the descent of F-splittings through normalization.","marker":"[Pos24]"},{"why":"Gives the abundance theorem for semi-log canonical surfaces in positive characteristic, allowing K_X ~_Q 0 in the proof of Theorem 6.3.","marker":"[Tan16]"},{"why":"Supplies the minimal model program and vanishing results for excellent surfaces used in the reductions to Mori fibre spaces and in the cohomology vanishings.","marker":"[Tan18]"},{"why":"Provides the connectedness principle for log canonical pairs used to control the number of genus-one components and boundary components in the index-one analysis.","marker":"[Pro01]"},{"why":"Supplies the foundational adjunction, different, conductor, and singularity theory that the entire reduction to the normalization pair relies on.","marker":"[Kol13]"}],"fun_headline_variants":["F-split K-trivial surfaces lift over Witt vectors","Equisingular char 0 lifts for F-split surfaces","From p to 0: equisingular lifts of singular surfaces","Witt vector lifts exist for F-split K-trivial surfaces","Singular F-split surfaces deform equisingularly"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof leans on a classification of the double-cover surfaces that arise when a rational boundary curve carries four singularities; if some globally F-split surface in characteristic p>2 produced a cover outside the listed types, the lifted gluing involution could fail to exist.","fun_headline_variants_meta":{"raw":{"variants":["F-split K-trivial surfaces lift over Witt vectors","Equisingular char 0 lifts for F-split surfaces","From p to 0: equisingular lifts of singular surfaces","Witt vector lifts exist for F-split K-trivial surfaces","Singular F-split surfaces deform equisingularly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0004,"raw_usage":{"total_tokens":1982,"prompt_tokens":730,"completion_tokens":1252,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":346,"completion_tokens_details":{"reasoning_tokens":1166}},"tokens_in":346,"tokens_out":1252,"duration_ms":10608,"temperature":1.0,"reasoning_tokens":1166,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:53:33.173445+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Look for a projective globally F-split slc surface X over an algebraically closed field of characteristic p>2 with K_X ≡ 0 whose normalization pair has a rational boundary component with four A1-singularities, and compute the μ2-equivariant minimal model of its index-one cover: if that model is not one of the five types in Lemma 7.7, or if the four branch points on the canonical lift of (\\mathbb{P}^1, \\frac{1}{2}\\sum q_i) carry no involution lifting τ, then Theorem 1.1 is false.","supporting_citations":[],"review_version":1}