{"id":"452bde9b-0efd-4fb6-a302-201e7258128a","arxiv_id":"2510.23959","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New 'logarithmic modification' topologies for fs log schemes are defined, with sheaf characterizations and a claimed correction to the full log étale site.","lead":"The paper introduces five new Grothendieck topologies—m-open, m-étale, m-smooth, m-fppf, and m-fpqc—for logarithmic schemes, treating log blow-ups as covers. It also claims to correct a flawed lemma in the standard full log étale topology, a correction that would affect many existing results in logarithmic geometry.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 1.1 claims log blow-ups are log modifications (universally surjective log étale monomorphisms), but standard log blow-ups such as Bl_0(A²)→A² are not monomorphisms; this invalidates Prop. 2.6 and the m-open topology's treatment of log blow-ups.","rationale":"The Reader's verdict focused on Definition 5.6 (the strengthened universal surjectivity) as the weakest assumption. My stress-test found a more fundamental problem: the paper's own Definition 1.1 of log modifications as monomorphisms is inconsistent with the claim that log blow-ups are log modifications. Standard log blow-ups, such as the blow-up of the rank-2 log point at the maximal ideal, have positive-dimensional fibers and are not monomorphisms in fsLogSch. This is not a subtle point of convention; it is a concrete, checkable failure of a central assertion. The proof of Proposition 2.6 explicitly uses the monomorphism property of log blow-ups, and the m-open topology is designed to treat log modifications as covers. If log blow-ups are not log modifications, then the m-open topology fails to include the principal examples, and the paper's proposed replacement for the log étale topology is not built on the objects it claims to cover. This is an internal inconsistency, not merely a disagreement with established literature. A specialist could verify the counterexample in a few lines of computation, making it a decisive test. Because the foundational definition is unsound as stated, the current version of the paper should be rejected, even though parts of the paper (e.g., the negative claim about the classical log étale site, if independently verified) might survive a rewrite.","tokens_in":21046,"tokens_out":49879,"duration_ms":484749,"concrete_test":"Let X = Spec(N² → C) be the standard log point and let Bl(X) be its log blow-up at the ideal (e₁,e₂). Explicitly compute the underlying scheme of Bl(X) and verify that it is P¹ with the toric log structure. Choose two distinct points p,q in the exceptional divisor that are not torus fixed, and let f,g: Spec(C) → Bl(X) be the resulting morphisms of fs log schemes, where Spec(C) carries the trivial log structure. Check that h∘f = h∘g for the blow-up map h: Bl(X) → X, but f ≠ g. This directly demonstrates that h is not a monomorphism, contradicting the paper's assertion that log blow-ups are log modifications. Additionally, consult [31, Theorem III.2.6.3(3)] and [33, Example 2.7] to confirm whether they actually claim monomorphism of log blow-ups.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's foundational Definition 1.1 defines a ‘logarithmic modification’ as a universally surjective log étale monomorphism, and then asserts that log blow-ups are log modifications, citing [33, Example 2.7] and [31, Theorem III.2.6.3(3)]. This assertion appears false. Consider X = Spec(N² → C), the standard rank-2 log point, and its log blow-up at the maximal ideal I = (e₁,e₂). The underlying scheme of Bl_I(X) is the exceptional divisor P¹. Let p and q be two distinct points in the exceptional divisor that are not torus fixed. Taking T = Spec(C) with the trivial log structure, the two scheme-theoretic inclusions T → P¹ at p and q define two distinct morphisms of fs log schemes T → Bl_I(X). Their compositions to X both factor through the unique point of X, hence are equal. Thus Bl_I(X) → X is not a monomorphism in fsLogSch. Since Bl_I(X) → X is universally surjective and log étale, it would be a log modification under Definition 1.1 if the monomorphism condition were met, but it is not. Therefore the paper's claim that log blow-ups are log modifications is internally inconsistent. This is not a mere terminological quibble: the proof of Proposition 2.6 relies on ‘log blow-ups are monomorphisms’ in an essential way (to conclude that base changes of open immersions are open immersions and that a surjective local isomorphism that is a monomorphism is an isomorphism). If log blow-ups are not monomorphisms, Proposition 2.6 collapses, and the m-open topology no longer contains the primary examples (log blow-ups) it was designed to treat as covers. The paper's central construction—the m-type topologies and their claimed correction of the log étale site—rests on this unsound foundation.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces new Grothendieck topologies (m-open, m-étale, m-smooth, m-fppf, m-fpqc) on the category of fs log schemes. The key move is to declare log modifications — defined in Definition 1.1 as universally surjective log étale monomorphisms — to be covers, alongside strict scheme-theoretic covers. The authors prove a chart criterion for m-open morphisms, an infinitesimal lifting criterion for m-étale and m-smooth morphisms (Theorem A), characterize sheaves on all m-type sites (Theorem B), and establish an equivalence between the m-open topos and Kato's valuative log space (Theorem D), with a resulting cohomological vanishing criterion. The central new result is Theorem C: in the standard full log étale site, a quasi-compact object must have all stalks of M^gp of rank at most 1, contradicting a lemma in [28]. The paper then proposes a strengthened notion of universal surjectivity (Definition 5.6) to repair the log étale topology and asserts that this fixes [28, Lemma 3.14].","tokens_in":21454,"tokens_out":33876,"duration_ms":381774,"significance":"If Theorem C is correct, this is a substantial correction to the literature on the full log étale topology, with consequences for several papers built on Nakayama's work. The proposed m-type topologies are a genuinely new framework that may be useful for log invariance phenomena. The paper is generally carefully written: proofs are detailed, dependencies are often explicitly flagged (e.g., Remark 2.7, reliance on Nizioł's resolution theorem), and the main theorems are stated in a falsifiable form. The global integralization theorem and the valuative-space description are valuable independent contributions. The main risk is the strong contradiction with [28], which is asserted with a fairly terse proof, and the foundational monomorphism property of log blow-ups, which is cited rather than proved in the text.","major_comments":[{"comment":"The definition of logarithmic modification and the proof of Proposition 2.6 hinge on the assertion that every log blow-up is a monomorphism in fsLogSch, citing [33, Example 2.7] and [31, Theorem III.2.6.3(3)]. This is non-obvious: scheme-theoretically, Bl_0(A²)→A² is not a monomorphism. A naive counterexample using two points in the exceptional divisor does not automatically lift to fsLogSch because log morphisms carry additional unit data; in an exact chart the local coordinate e1/e2 is a unit, so distinct points can induce different log maps to the base. Nevertheless, since Definition 1.1 and the m-open topology rest entirely on this point, please include a proof, or at least state precisely which result in the cited references asserts this and explain why the scheme-theoretic intuition is misleading. This is load-bearing for Proposition 2.6 and for the claim that log blow-ups are m-op","section":"§1.1, Definition 1.1; §2.1, Proposition 2.6"},{"comment":"The proof of Theorem C is quite terse at the point where a finite subcover of the valuative cover of X is used to conclude that each étale chart U_i admits a quasi-compact log étale cover by valuative fs log schemes. The intended argument is presumably that pulling back the finite subcover along U_i→X gives a finite cover of U_i, but this should be written out explicitly. More importantly, the proof relies on [28, Lemma 3.11] and [28, Proposition 3.7] even though the paper claims that [28] is largely broken by Theorem C. Please verify that these cited results are independent of the allegedly erroneous Lemma 3.14, or supply alternative proofs. This matters because Theorem C is the paper's main correction.","section":"§5.1, Proof of Theorem C"}],"minor_comments":[{"comment":"The phrase 'strict étale locally in T exactly one(at least one) dotted arrow' is missing a space/comma; also 'locally in T' is ambiguous — it means strict étale local on the thickening T.","section":"§1.2, Definition 1.4"},{"comment":"In the statement, 'formally étale (resp. m-smooth)' should presumably read 'formally étale (resp. formally smooth)' for consistency with the proof.","section":"§3.1, Lemma 3.2"},{"comment":"The proof refers to Theorem 6.3 and Lemma 6.5 before they are stated. Add forward references or move the needed preliminaries earlier.","section":"§4, Proposition 4.3"},{"comment":"'form a topology base' should be 'form a base for the topology'.","section":"§6.1, Theorem D proof"},{"comment":"The statements that Theorem C 'seems to break most of [28]' and 'impacts virtually every paper' are stronger than what is proved. I suggest tempering these claims or spelling out which specific results in [28] fail.","section":"§1.4, Theorem C discussion"},{"comment":"The equivalence of the three conditions in Lemma 5.4 is stated for 'an integral log scheme T' but the fiber products are taken in different categories (integral vs saturated). The notation in condition (3) is also a bit compressed; a short clarification would help.","section":"§5.2, Lemma 5.4"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about Definition 1.1 does not, on careful reading, land: the proposed counterexample with two points in the exceptional fiber of a log blow-up does not produce equalized morphisms in fsLogSch because the log structure records the unit coordinate. I do not see a fatal internal inconsistency here. The main risk is Theorem C, which contradicts a published lemma in [28] and whose proof is terse at a crucial point. The paper should be sent back for a careful expansion of that proof and for an explicit proof of the monomorphism property of log blow-ups. The novelty is potentially significant, so major revision seems right."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a serious paper, more careful than most, but its headline result (Theorem C) contradicts Nakayama and I couldn't verify it; the m-topology framework is the real contribution and looks workable. The stress-test note about log blow-ups failing to be monomorphisms is, as far as I can tell, wrong.\n\nWhat's actually new: the m-open, m-étale, m-smooth, m-fppf, and m-fpqc topologies, with Theorem A giving a clean lifting criterion for m-étale/m-smooth, and Theorem D identifying the m-open topos with the topos of Kato's valuative space. The sheaf characterization in Theorem B is also useful, and Theorem E's explicit formula for log dimension is a nice concrete payoff. The paper is transparent about its dependencies — Remark 2.7 openly says that Proposition 2.6 needs the strengthened notion of universal surjectivity, and the proof of Theorem C is laid out in enough detail to be checkable.\n\nThe soft spots are real. First, Theorem C is a genuinely bold claim: it says the usual full log étale site has quasi-compact objects only when all stalks of M^gp have rank ≤ 1, which would invalidate Nakayama's Lemma 3.14 and much of [28]. The proof is plausible, but such a strong correction to a published framework needs independent verification before I'd rely on it. Second, the whole repair depends on the nonstandard Definition 5.6 of universal surjectivity (base change along all integral log schemes). If that notion is wrong or too restrictive, the quasi-compactness results and the m-type coverings lose their footing; the paper does give evidence it is well-behaved (Proposition 5.7), but it is a pivot point.\n\nOn the stress-test: the claim that Bl_0(A²)→A² is not a monomorphism because two trivial log points on the exceptional divisor define distinct morphisms is mistaken. Those are not morphisms of log schemes: at a point on the exceptional divisor, the log structure has the exceptional divisor as a non-unit generator, so it does not pull back to the trivial log structure. The two scheme-theoretic inclusions do not lift to fsLogSch. So that particular objection collapses.\n\nWho should read this: anyone working on log étale cohomology, log motives, or log moduli. The m-topologies are a plausible working substitute if Theorem C holds, and even if Theorem C is wrong, the m-framework is a well-built variant worth knowing. It deserves a serious referee, with particular attention to Theorem C and to how far Definition 5.6 changes the meaning of \"log étale cover.\"","headline":"Bold, mostly careful paper introducing m-topologies and claiming a fix to the full log étale site; Theorem C needs a specialist check, but the stress-test about log blow-ups not being monomorphisms does not survive contact with the log structures.","tokens_in":21979,"tokens_out":5886,"would_cite":true,"duration_ms":60753,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14A21"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper argues that the standard log étale topology is subtly broken and that m-type topologies, built by treating log blow-ups as covers, are the correct fix.","keywords":["logarithmic geometry","Grothendieck topology","log modifications","m-étale morphisms","log étale site","valuative log space","universal surjectivity","log blow-up"],"falsifier":"Take an fs log scheme X with a point whose stalk M^gp_{X,x} has rank ≥ 2 and exhibit a single quasi-compact log étale cover (finite union) covering X; Theorem C says this is impossible, so a concrete such cover would disprove it. Alternatively, a log blow-up whose base change along some integral (non-fs) log scheme is not surjective would falsify Proposition 5.7(3) and the new universal surjectivity definition.","tokens_in":20884,"feed_emoji":"🪵","tokens_out":7246,"duration_ms":65396,"temperature":0.7,"pith_summary":"Logarithmic invariants in geometry often survive log blow-ups, so a good topology for log schemes should treat log modifications as covers. This paper introduces m-type topologies—m-open, m-étale, m-smooth, m-fppf, and m-fpqc—in which log modifications, defined as universally surjective log étale monomorphisms, become covering maps. Along the way it identifies an error in the standard full log étale topology: a quasi-compact object in that site must have rank-≤1 log-characteristic stalks, which is far too restrictive and contradicts a published lemma; the paper repairs this by strengthening the definition of universal surjectivity so that it is tested against all integral log schemes. With that repair, the m-open topos of an fs log scheme is equivalent to the topos of its valuative log space, and sheaves on the m-type sites admit a simple characterization: descend along strict covers and along log blow-ups.","feed_headline":"Log blow-ups become covers: repairing the log étale site","feed_subtitle":"A stronger definition of universal surjectivity restores quasi-compact objects and a cohomology vanishing bound.","key_machinery":"The load-bearing objects are m-type morphisms and the strengthened notion of universal surjectivity. A log modification is declared to be a universally surjective log étale monomorphism, which automatically becomes proper under the new definition; m-étale morphisms are then defined by a chart criterion (or equivalently by a formal lifting criterion against arbitrary exact first-order thickenings). The technical core is Definition 5.6: a morphism of fs log schemes is universally surjective only if its base change along every integral log scheme is surjective, a strictly stronger requirement than the usual one. This redefinition makes the m-type sites functorial and makes log blow-ups into cov","core_discovery":"The central claim is that the standard full log étale topology is subtly broken and that a family of m-type topologies, with a corrected notion of universal surjectivity, is the right replacement. Concretely: in the small or big log étale site of an fs log scheme, an object is quasi-compact exactly when its underlying scheme is quasi-compact and every stalk of M^gp has rank ≤ 1; this contradicts a lemma in the published literature and implies the log étale site has too few quasi-compact objects, so results built on that lemma need revisiting. The paper's fix is to define universal surjectivity as surjectivity after all base changes along integral log schemes; under this definition log modifi","pith_inferences":["If the strengthened universal surjectivity is accepted, the m-type topologies likely agree with the 'dividing' topologies introduced in the motivic literature, giving a direct categorical foundation for logarithmic motives.","The redefinition effectively says that fs log schemes are not the correct test category for surjectivity; the broader category of integral log schemes is. This suggests that other log-geometric notions (e.g., properness, flatness) may also need to be reformulated with integral test objects.","The valuative-space equivalence suggests a new computational route for log étale cohomology: compute on the smaller, spectral valuative space rather than on the log scheme itself.","The global integralization theorem—every fs log scheme becomes Zariski-log after a log blow-up—could simplify many reduction arguments if it holds for the strengthened notion, since Zariski log structures are far easier to chart."],"forward_implications":["The standard full log étale site has far fewer quasi-compact objects than previously believed, so the many papers using the classical log étale topology will need to check their quasi-compactness and descent arguments.","Any invariant that is invariant under log blow-ups automatically becomes a sheaf on the m-type sites; in particular log Gromov–Witten, log Chow, and log Hilbert theories all descend.","A presheaf is a sheaf on an m-type site if and only if it satisfies descent for strict type-covers and maps log blow-ups to isomorphisms—a drastically simplified sheaf condition.","Cohomology of any abelian sheaf on the m-open site vanishes above the logarithmic dimension, which is computed by an explicit formula involving point dimensions and log stalk ranks.","The m-open topos equivalence with the valuative log space topos gives a concrete spectral space model for the m-open site, enabling topological dimension arguments."],"fun_headline_variants":["New m-topologies fix log étale site","Log blow-ups as covers: m-étale repair","Correcting log étale: m-open, m-étale, m-smooth","Integralization by blow-ups: m-open topos","m-topologies: log modifications become covering"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire correction rests on the strengthened definition of universal surjectivity (Definition 5.6); if a log blow-up fails to be surjective after base change to some integral log scheme, or if this definition is too restrictive for the claimed proofs (the paper itself notes Proposition 2.6 depends on it), the repair of the log étale topology collapses, as does any reliance on the resolution theorem that reduces to Zariski log structures.","fun_headline_variants_meta":{"raw":{"variants":["New m-topologies fix log étale site","Log blow-ups as covers: m-étale repair","Correcting log étale: m-open, m-étale, m-smooth","Integralization by blow-ups: m-open topos","m-topologies: log modifications become covering"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000227,"raw_usage":{"total_tokens":1280,"prompt_tokens":690,"completion_tokens":590,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":434,"completion_tokens_details":{"reasoning_tokens":508}},"tokens_in":434,"tokens_out":590,"duration_ms":5641,"temperature":1.0,"reasoning_tokens":508,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T07:50:08.797306+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an fs log scheme X with a point whose stalk M^gp_{X,x} has rank ≥ 2 and exhibit a single quasi-compact log étale cover (finite union) covering X; Theorem C says this is impossible, so a concrete such cover would disprove it. Alternatively, a log blow-up whose base change along some integral (non-fs) log scheme is not surjective would falsify Proposition 5.7(3) and the new universal surjectivity definition.","supporting_citations":[],"review_version":1}