{"id":"fde793fe-872f-4b85-8bfd-a1848aae547b","arxiv_id":"2608.01441","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Maximal-variation families of general-type varieties over smooth proper DM stacks with generically trivial stabilizers force the log canonical bundle K_X + Delta to be big.","lead":"This paper proves that a smooth proper Deligne-Mumford stack which carries a maximal-variation family of varieties of general type has a big log canonical bundle, meaning the stack is of log general type. It extends Popa-Schnell's theorem from ordinary varieties to stacks, yielding sharper statements about the coarse moduli space.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Bigness conclusion is imported from unpublished prequels; the decisive link is [CMZ26b, Thm. 5.1] producing a logarithmic Higgs bundle with F^0 big, which §4 of this paper does not prove.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing concern: the proof of Theorem A depends on unpublished, self-cited prequels, especially [CMZ26b, Thm. 5.1] and [CMZ26d, Thm. B]. My read of the paper confirms this. The paper's own new contribution—Proposition 3.2, 3.4, and the duality lemmas in §2—is substantial and appears internally coherent, and the examples in §5 are reasonable applications if Theorem A holds. However, the final bigness statement is not derived in this paper; it is imported. The concern is not that the prequels are wrong, but that the central claim cannot be fully assessed from this text alone. That is precisely a conditional, not a rejection. I therefore recommend keeping the reader's CONDITIONAL verdict. The proposed concrete test targets the most delicate point in the prequel chain: the log poles of the Higgs bundle must be controlled with respect to Δ, not just with respect to the larger singular support E, because the Viehweg–Zuo sheaf construction in [CMZ26d, Thm. B] requires exactly that. No ad hominem is intended; the critique is structural and verifiable.","tokens_in":34885,"tokens_out":16377,"duration_ms":155461,"concrete_test":"Independently verify [CMZ26b, Thm. 5.1] in the explicit example X = [P^1/S_3] with Δ = 1/2 p_∞ from the introduction. Run the cyclic-cover construction of Proposition 3.4 and check that the resulting graded sheaves F^• satisfy θ^p(F^p) ⊆ Ω^1_X(log Δ)⊗F^{p+1} with F^0 a big line bundle, not merely Ω^1_X(log E) for a larger divisor E. If the construction only gives logarithmic poles along E = Δ∪S, then [CMZ26d, Thm. B] cannot be applied to produce H ⊆ (Ω^1_X(log Δ))^{⊗s}, and the proof of Theorem A collapses at this step. If it succeeds for this case, the decisive prequel link is at least locally consistent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem A is established only as a reduction to a chain of four self-cited unpublished preprints. The proof in §4 proceeds: after Proposition 3.2 and 3.4, the paper invokes [CMZ26b, Thm. 4.13] to obtain a Hodge module and then [CMZ26b, Thm. 5.1] to obtain a graded logarithmic Higgs bundle E^• with F^• ⊆ E^•, θ^p(F^p) ⊆ Ω^1_X(log Δ)⊗F^{p+1}, and F^0 a big line bundle. From that data, [CMZ26d, Thm. B] is used to produce a Viehweg–Zuo sheaf H with big determinant, and [CMZ26a, Cor. B] gives bigness of K_X+Δ. The final bigness conclusion therefore rests entirely on the correctness of these prequels, especially on [CMZ26b, Thm. 5.1]. The paper itself flags that the generically-trivial-stabilizer hypothesis is only used in Proposition 3.2 and expects it can be removed; this is an acknowledged limitation, not a gap. But the more load-bearing issue is that the passage from a Hodge module to a logarithmic Higgs bundle with poles exactly along Δ, and then to a Viehweg–Zuo sheaf, is not demonstrated in this text. If [CMZ26b, Thm. 5.1] only yields a Higgs field with θ(F^p) ⊆ Ω^1_X(log E) for E = Δ∪S, rather than Ω^1_X(log Δ), then the inclusion H ⊆ (Ω^1_X(log Δ))^{⊗s} is unjustified and Theorem A does not follow. The reader's CONDITIONAL verdict is therefore appropriate; the central claim is plausible but not yet independently verified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem A, a Deligne-Mumford stack version of Viehweg--Popa--Schnell hyperbolicity: if f: Y -> X is a surjective schematic projective morphism of smooth proper integral DM stacks with projective coarse spaces, connected fibers, maximal variation, and geometric generic fiber of general type or admitting a good minimal model, then for any reduced divisor Δ containing the discriminant, K_X + Δ is big, provided X has generically trivial stabilizers. The proof follows the Popa--Schnell strategy: mild reduction, fibered-product and cyclic-cover constructions, construction of a Hodge module and logarithmic Higgs bundle with a big piece F^0, then a Viehweg--Zuo sheaf, and finally bigness via a Campana--Păun-type theorem. Corollary B states that the corresponding pair on the coarse moduli space is of log general type. Applications to moduli of curves, admissible covers, abelian varieties, and cubic surfaces are given. The paper is structured as a reduction to four unpublished preprints by the same authors ([CMZ26a--d]) and to an unpublished manuscript of Rydh; most of the genuinely new material is in §2--§3, while the decisive Hodge-theoretic steps in §4 are quoted from those preprints.","tokens_in":35411,"tokens_out":5048,"duration_ms":49956,"significance":"If the result is correct, it is a substantial extension of a landmark theorem: it gives bigness of the logarithmic canonical bundle directly on moduli stacks, not only on their coarse spaces, and it yields sharper boundary coefficients on coarse spaces. The applications in §5 are meaningful and well chosen, particularly the derivation of K_{M_g}+δ_0 big for g≥2 and the discussion of the second Voronoi compactification. The paper also contains useful extensions of duality, root-stack covering tricks, and mild reduction to DM stacks. However, the proof's central chain relies on a series of unpublished, same-author preprints and one unpublished personal manuscript, and the exact polar-locus statement needed from [CMZ26b, Thm. 5.1] is asserted but not proved in this text. These dependencies make the current version unsuitable for acceptance without substantial verification.","major_comments":[{"comment":"The final bigness conclusion is not proved in this paper; it is imported from [CMZ26b, Thm. 4.13 and Thm. 5.1], [CMZ26d, Thm. B], and [CMZ26a, Cor. B]. After Proposition 3.2 and Proposition 3.4, the proof consists of checking that the hypotheses of [CMZ26b, §4.1.1--3] hold and then invoking these results. Since all four are unpublished preprints by the same authors, the central claim of Theorem A is established only conditional on their correctness. This is not, in my reading, a circularity, but it is a verification gap: a referee of this paper cannot check the decisive step from a Hodge module to a Viehweg--Zuo sheaf. I ask that the authors either include the statements (and enough proof) of the specific results used, or provide evidence that the preprints have been refereed and accepted.","section":"§4, proof of Theorem A"},{"comment":"The proof requires a logarithmic Higgs bundle with θ^p(F^p) ⊆ Ω^1_X(log Δ) ⊗ F^{p+1}, i.e. with poles only along the discriminant divisor Δ. The text asserts this immediately after invoking [CMZ26b, Thm. 5.1]. But the preceding construction also produces the singular support S of the Hodge module M, and the natural output would be a Higgs field with poles along E = Δ ∪ S. If the theorem only yields poles along E, the subsequent inclusion H ⊆ (Ω^1_X(log Δ))^{⊗s} is unjustified and Theorem A does not follow. The paper needs to state the precise theorem from [CMZ26b] and explain why the poles are exactly along Δ, or give a proof of this key reduction.","section":"§4, paragraph beginning 'We may therefore employ [CMZ26b, Thm. 5.1]'"},{"comment":"Mild reduction for DM stacks is load-bearing: it feeds into Propositions 3.2 and 3.4, and hence into the whole proof. In the proof of Proposition 1.5, Raynaud--Gruson flattening for stacks is invoked from [Ryd11, Thm. A], an unpublished personal manuscript (cited as a 2011 PDF). The argument says 'the proof is identical to [VZ03, §2]' and replaces [AK00] and Raynaud--Gruson by [Ryd11]. This is a load-bearing external dependency. The authors should either provide a complete proof of the flattening step for the specific morphism X'_0 -> X, or cite a published version with precisely stated hypotheses. The current citation is not sufficient for a journal referee to verify the reduction.","section":"Proposition 1.5"},{"comment":"Lemma 2.8(iv) is used to obtain the crucial isomorphism (3.3) in Proposition 3.2, which underlies the global generation argument. The proof of Lemma 2.8(iv) is a long reduction to curves, and in Step 3 the claim that α is an isomorphism for very general hyperplanes H_i is justified by a Tor-vanishing argument and the isomorphism (2.15). However, the passage from the isomorphism (2.15) to the equality of determinants needed for (3.3) is not fully spelled out, and the role of [Kol23, 2.68.2] is quoted without stating the required conditions. Since this lemma is part of the new content of the paper and is used essentially in the construction of the Viehweg--Zuo sheaf, the proof should be expanded or the missing hypotheses stated.","section":"Lemma 2.8(iv) and Proposition 3.2, Step 1"}],"minor_comments":[{"comment":"The paper repeatedly says 'the proof is identical to [VZ03, §2]' or 'follows étale-locally from the case of varieties.' While often acceptable, several of these reductions are not completely formal because the objects are stacks; the authors should at least point to the precise statement being generalized. This is a clarity issue, not a correctness issue.","section":"Throughout"},{"comment":"The diagram contains repeated entries 'X′ X′ X′ X′' and the spacing makes it hard to read. Please format the commutative diagrams more carefully; the same problem appears in (2.9) and (3.4).","section":"§1.5, diagram (1.3)"},{"comment":"The notation 'A_X^◦' and 'A^◦_X^◦' in the discussion after [CMZ26b, Thm. 4.13] is confusing; A and its restriction to X^◦ should be defined explicitly. Also, the role of the divisor S' in [CMZ26b, §4.1.3] is only sketched; please make the comparison precise.","section":"§4, proof of Theorem A"},{"comment":"In the construction of β, the sentence 'the pull back of the trace factors through the co-unit of adjunction' is terse. Since the argument is central to the inclusion (2.10), consider expanding the justification that the kernel of the co-unit is torsion and that β is injective.","section":"§2.2.2, Lemma 2.6"},{"comment":"The coefficients 1/2, 5/6, and 1/3 appear without a full derivation; please add a sentence explaining where each coefficient comes from, especially the 5/6 in (5.3).","section":"§5.4, equations (5.2)--(5.3)"},{"comment":"[Ryd11] is cited as an unpublished manuscript; [Nir09] is cited as an arXiv preprint. If published versions exist, they should be cited. For the four preprints [CMZ26a--d], arXiv numbers are given, which is helpful, but it would be even better to state their status (submitted/in press).","section":"References"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the manuscript's proof is a chain of reductions to four unpublished preprints by the same authors, and one unpublished personal manuscript. The reader's conditional verdict is appropriate. I would encourage the editor to ask the authors to make the key statements from [CMZ26b, Thm. 5.1] and [CMZ26d, Thm. B] available in full, or to have the prequels refereed before this paper is accepted. The paper itself is clearly written and the examples are valuable, but the central bigness claim cannot be independently verified from the present text."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this paper states the right stack-level generalization of Popa–Schnell, and the statement is genuinely useful, but the proof of Theorem A is almost entirely delegated to four unpublished preprints by the same authors. You cannot assess the main theorem from this text alone.\n\nWhat's new: Theorem A extends [PS17] to smooth proper DM stacks with generically trivial stabilizers, and Corollary B gives log general type of the coarse pair with a sharper boundary divisor. The examples are a real strength—especially the elliptic curve case and the cubic surfaces discussion, where the stack-level result visibly beats applying [PS17] to the coarse space. Sections 1–2 also contain genuinely useful stack versions of mild reduction and relative duality; those look careful.\n\nThe soft spot is structural. In §4, after Proposition 3.2 and 3.4, the proof invokes [CMZ26b, Thm. 4.13 and 5.1] to get the graded logarithmic Higgs bundle with F^0 big, then [CMZ26d, Thm. B] and [CMZ26a, Cor. B] to get a Viehweg–Zuo sheaf and bigness. That is the core of the argument, and none of it is shown here. The stress-test's worry about [CMZ26b, Thm. 5.1] only giving poles along E = Δ∪S rather than along Δ is legitimate; if that's the case, the inclusion H ⊆ (Ω^1_X(log Δ))^{⊗s} is unjustified and Theorem A doesn't follow. Proposition 1.5 also relies on [Ryd11], an unpublished note. The authors are honest about the dependencies, so this is not an internal flaw, just a lack of self-containment.\n\nWho's this for? Specialists in moduli theory and birational geometry; others will drown in stack machinery. I'd send it to peer review, but the editor should require the full prequel series to be available and ask referees to verify the chain, especially [CMZ26b, Thm. 5.1]. If that holds up, this is a solid contribution. For now, treat Theorem A as conditionally established in the series, not in this paper alone.","headline":"Plausible stack-level Viehweg hyperbolicity, but the proof is in four unpublished prequels; judge the whole series, not just this paper.","tokens_in":35801,"tokens_out":4058,"would_cite":false,"duration_ms":37355,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14D23","14J10","14D07","14E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a smooth proper Deligne–Mumford stack which is the base of a maximal-variation family of varieties of general type has big log canonical bundle, making the stack itself of log general type.","keywords":["Deligne–Mumford stacks","log general type","big line bundles","Viehweg hyperbolicity","moduli of varieties","Hodge modules","logarithmic Higgs bundles","KSBA moduli"],"falsifier":"Check the four cited preprints at the exact points used in Section 4—[CMZ26b, Thm. 4.13 and Thm. 5.1], [CMZ26d, Thm. B], and [CMZ26a, Cor. B]—for a counterexample satisfying the hypotheses of Theorem A; any failure there would sink the proof. Independently, compute the class of $K_X+\\Delta$ on the elliptic-family quotient stack $[\\mathbb{P}^1/S_3]$ from the introduction: the paper says a positive power descends to the class $\\frac{13}{6}p_0$ on $\\mathbb{P}^1$, which is big; a computation showing that class is not big would refute the theorem's stack-level claim.","tokens_in":34815,"feed_emoji":"📐","tokens_out":16337,"duration_ms":135349,"temperature":0.7,"pith_summary":"This paper aims to show that a moduli stack that carries a maximal-variation family of varieties of general type is itself of log general type, not just its coarse moduli space. The main theorem states that for a smooth proper Deligne–Mumford stack X over C with projective coarse moduli space and generically trivial stabilizers, any surjective projective family Y→X with connected fibers, geometric generic fiber of general type (or admitting a good minimal model), and maximal variation has big log canonical bundle $K_X+\\Delta$ for every reduced divisor $\\Delta$ containing the discriminant. Big here means a positive tensor power descends to a big line bundle on the coarse space. The paper's corollary is that the coarse pair $(X,R+\\Delta)$, where $R$ is the ramification $\\mathbb{Q}$-divisor of the coarse morphism, is of log general type; this is sharper than the previously known variety-level result, because $R+\\Delta$ has smaller coefficients than its round-up. A reader should care because the stack-level formulation is the natural one for moduli problems, and the paper shows it yields explicit divisor-class consequences for moduli of curves, abelian varieties, and cubic surfaces.","feed_headline":"General-type families make the moduli stack itself log general type","feed_subtitle":"A new theorem extends hyperbolicity to the Deligne–Mumford stacks themselves, sharpening the coarse-space result.","key_machinery":"The load-bearing object is the Viehweg–Zuo sheaf: a saturated coherent subsheaf $H$ of a tensor power of the logarithmic cotangent bundle, $(\\Omega^1_X(\\log\\Delta))^{\\otimes s}$, whose determinant is big, sitting in a short exact sequence $0\\to H\\to(\\Omega^1_X(\\log\\Delta))^{\\otimes s}\\to Q\\to0$. Finding such a sheaf is enough, because one then knows $\\det Q$ is pseudo-effective and a positive multiple of $K_X+\\Delta$ is 'big plus pseudo-effective', hence big. The proof builds the sheaf by first using mild reduction, fibered products and cyclic covers (realized through root stacks) to make the variation of the family sufficiently positive; then a Hodge-module/logarithmic-Higgs-bundle construc","core_discovery":"The paper's central claim is Theorem A: let $f:Y\\to X$ be a surjective schematic projective morphism of smooth proper integral Deligne–Mumford stacks over $\\mathbb{C}$ with projective coarse moduli spaces and connected fibers, and let $\\Delta$ be any reduced divisor containing the discriminant of $f$. If the geometric generic fiber is of general type or admits a good minimal model, if $f$ has maximal variation, and if $X$ has generically trivial stabilizers, then $K_X+\\Delta$ is big. This is a direct generalization of the known variety-level hyperbolicity theorem [PS17] to stacks. The discovery is that the stack itself, rather than only its coarse space, is of log general type: bigness is te","pith_inferences":["Beyond the paper: the generically-trivial-stabilizer hypothesis is used in one global-generation step and the authors expect it can be dropped; if that expectation is borne out, the theorem would apply directly to stacks such as toroidal compactifications with generic $\\mathbb{Z}_2$ automorphisms, without passing to a finite cover.","Beyond the paper: because the proof is a reduction to four companion preprints, an independent verification could be made by writing out the constructed Viehweg–Zuo sheaf in a concrete example, such as the elliptic-curve quotient stack, which would test the whole chain without relying on the preprints.","Beyond the paper: the sharper boundary coefficients in Corollary B suggest a general principle for moduli stacks—stabilizers contribute their ramification divisor with coefficients below 1, so the stack's own log canonical bundle is a more economical witness of hyperbolicity than the coarse log canonical bundle with rounded boundary.","Beyond the paper: the same strategy may apply to other KSBA moduli stacks of stable log pairs, where the discriminant is naturally a stack divisor; the theorem gives a template for proving log general type before a coarse-space family has been constructed."],"forward_implications":["The moduli stack itself, not just its coarse space, becomes the subject of hyperbolicity: under the theorem's hypotheses, $K_X+\\Delta$ is big, so the stack is of log general type.","The coarse-space corollary is the log general type of $(X,R+\\Delta)$, with $R$ the ramification $\\mathbb{Q}$-divisor of the coarse morphism; this gives a sharper boundary than adding $\\lceil R+\\Delta\\rceil$, since $R+\\Delta$ has smaller coefficients.","For moduli of stable curves, the theorem recovers $K_{\\mathcal{M}_g}+\\delta_0$ big for $g\\ge2$, and gives explicit coarse-space statements such as $K_{M_3}+\\frac12 H+\\frac12\\Delta_1+\\Delta_0$ big.","For the second Voronoi compactification of the moduli of abelian varieties, $K+\\Delta$ is big for every $g$, and it is not nef for $g\\ge3$ while being the pullback of an ample class for $g\\le2$.","For the KSBA moduli space of cubic surfaces, different geometric routes yield bigness of $K_{M_{\\text{cub}}}+\\Delta$ with different boundary coefficients, and the paper locates these in the nef and ample cones."],"supporting_citations":[{"why":"The theorem being generalized; its strategy—Viehweg–Zuo sheaves plus Hodge-module/Higgs-bundle construction—is followed throughout.","marker":"[PS17]"},{"why":"Supplies the positivity criterion that the determinant of the quotient $Q$ is pseudo-effective, used together with the stack analogue in the final bigness step.","marker":"[CP19]"},{"why":"Provides the stack-level pseudo-effectivity statement and the criterion that a big plus pseudo-effective line bundle on a DM stack is big; both are used in the final step of Theorem A.","marker":"[CMZ26a]"},{"why":"Prequel with the Hodge-module and logarithmic-Higgs-bundle constructions (Thm. 4.13, Thm. 5.1) and the log-resolution reduction (Lem. 1.9) used in the proof of Theorem A.","marker":"[CMZ26b]"},{"why":"Thm. B: from a logarithmic Higgs bundle with a big lowest graded piece, produces a Viehweg–Zuo sheaf with big determinant on a DM stack.","marker":"[CMZ26d]"},{"why":"Develops mild reduction and the fibered-product/cyclic-cover construction for varieties, which the paper extends to stacks in Propositions 1.5 and Lemmas 2.8–2.9.","marker":"[VZ03]"},{"why":"Provides the weak positivity, relative duality, and pluricanonical push-forward lemmas that Lemma 2.6 and Lemma 2.7 generalize to DM stacks.","marker":"[Vie83]"},{"why":"Prop. A.1/A.2: the geometric construction ensuring global generation of relative pluricanonical bundles after base change, used in Proposition 3.2.","marker":"[PTW19]"},{"why":"Thm. A: Raynaud–Gruson flattening for DM stacks, cited in the proof of the mild-reduction Proposition 1.5; the paper lists it as an unpublished manuscript.","marker":"[Ryd11]"},{"why":"Weak semi-stable reduction in characteristic zero, used through Proposition 1.2 as the starting point for mild reduction.","marker":"[AK00]"}],"fun_headline_variants":["General-type families make the moduli stack log general type","Moduli stacks of general type families are log general type","General type variation: the stack itself is log general type"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The proof of Theorem A relies on the correctness of four companion preprints cited in Section 4 and on an unpublished flattening result used in Proposition 1.5; if any of these has a gap, the chain that produces the Viehweg–Zuo sheaf and concludes bigness is broken.","fun_headline_variants_meta":{"raw":{"variants":["General-type families make the moduli stack log general type","Moduli stacks of general type families are log general type","General type variation: the stack itself is log general type"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000247,"raw_usage":{"total_tokens":1342,"prompt_tokens":666,"completion_tokens":676,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":410,"completion_tokens_details":{"reasoning_tokens":623}},"tokens_in":410,"tokens_out":676,"duration_ms":6569,"temperature":1.0,"reasoning_tokens":623,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T00:09:10.967855+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the four cited preprints at the exact points used in Section 4—[CMZ26b, Thm. 4.13 and Thm. 5.1], [CMZ26d, Thm. B], and [CMZ26a, Cor. B]—for a counterexample satisfying the hypotheses of Theorem A; any failure there would sink the proof. Independently, compute the class of $K_X+\\Delta$ on the elliptic-family quotient stack $[\\mathbb{P}^1/S_3]$ from the introduction: the paper says a positive power descends to the class $\\frac{13}{6}p_0$ on $\\mathbb{P}^1$, which is big; a computation showing that class is not big would refute the theorem's stack-level claim.","supporting_citations":[],"review_version":1}