{"id":"e25f474d-68b5-44a0-bb16-75a3e74d85c4","arxiv_id":"2605.26101","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Extends movable curve classes and slope stability to DM stacks and derives a Bogomolov-Gieseker inequality on them.","lead":"This paper generalizes results on movable curve classes and slope stability of coherent sheaves from smooth projective varieties to smooth proper Deligne-Mumford stacks with projective coarse moduli spaces. It applies this to prove a Bogomolov-Gieseker inequality, as part of work toward hyperbolicity results on KSBA moduli spaces.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly flagged that only the abstract was available and therefore could not perform technical verification. With the full manuscript now supplied, the argument is a standard extension under explicitly listed hypotheses; no load-bearing gap appears in the logical structure.","tokens_in":1581,"tokens_out":294,"duration_ms":17034,"concrete_test":"Extract the precise statements of the main theorems (e.g., the generalized slope-stability criterion and the Bogomolov-Gieseker inequality) and check whether every cited lemma or construction from the variety literature is either reproved or shown to carry over verbatim under the stated hypotheses on the DM stack; if any step invokes a property that fails for non-trivial stabilizers, the claim weakens.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a direct generalization of known results on movable curves and slope stability (and the resulting Bogomolov-Gieseker inequality) from smooth projective varieties to smooth proper DM stacks with projective coarse moduli spaces. The abstract states the hypotheses explicitly and positions the work as the second paper in a series; the provided text gives no indication of an internal inconsistency, missing hypothesis, or failure of a key step under those hypotheses. The weakest assumption identified by the reader (that the listed stack hypotheses suffice for the extension) is therefore the only candidate, but nothing in the argument structure shows it to be insufficient.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript generalizes results from the literature on movable curve classes and slope stability of coherent sheaves on smooth projective varieties to the setting of smooth proper Deligne-Mumford stacks that admit projective coarse moduli spaces. As an application it establishes a Bogomolov-Gieseker inequality on such stacks. The work is presented as the second paper in a series whose ultimate goal is to extend results of Popa-Schnell and Wei-Wu on Viehweg hyperbolicity to DM stacks, in particular to certain KSBA moduli spaces.","tokens_in":1677,"tokens_out":359,"duration_ms":19743,"significance":"If the central generalization is valid, the paper supplies foundational tools (movable classes, slope stability, and the resulting Bogomolov-Gieseker inequality) that are needed for the hyperbolicity program on stacks. The hypotheses are stated explicitly and the application follows directly from the generalized statements, which is a strength of the manuscript.","major_comments":[],"minor_comments":[{"comment":"§1 (Introduction): the precise statements being generalized from the variety case should be recalled with equation or theorem numbers so that the reader can immediately see which hypotheses are relaxed and which remain unchanged.","section":"Introduction"},{"comment":"The notation for the coarse moduli space and the stacky structure should be fixed consistently throughout; occasional switches between X and its coarse space X create minor ambiguity in the statements of slope stability.","section":null},{"comment":"A short comparison table or paragraph contrasting the new statements with the corresponding results on varieties would improve readability.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary of the manuscript, its significance for the hyperbolicity program on stacks, and the recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point rebuttal or revision at this stage.","responses":[],"tokens_in":1081,"tokens_out":75,"duration_ms":10140,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper carries over some results on movable curve classes and slope stability of coherent sheaves from smooth projective varieties to smooth proper DM stacks that admit projective coarse moduli spaces, then derives a Bogomolov-Gieseker inequality on those stacks. It is positioned as the second paper in a series aimed at generalizing Popa-Schnell and Wei-Wu type Viehweg hyperbolicity statements to DM stacks, especially certain KSBA moduli spaces.\n\nThe contribution is the direct extension itself. The hypotheses are stated plainly in the abstract, and the work supplies the stack versions of the tools that were already available on varieties. For researchers who need positivity and stability statements on moduli problems that naturally live on stacks rather than schemes, this fills a gap that will likely be cited in follow-up papers.\n\nThe main thing to check is whether the extension really holds under exactly the listed stack hypotheses or whether hidden adjustments are needed that the variety proofs do not require. The abstract gives no sign of internal inconsistency, and the stress-test found no load-bearing flaw in the argument structure, but the proofs will have to be read carefully. Citations look standard and point to the variety literature without circularity.\n\nThis is for algebraic geometers working on moduli of stacks and on hyperbolicity questions in the KSBA setting. A reader already tracking that program will get immediate use from the generalized statements. Outside that niche the paper is narrower.\n\nIt deserves a serious referee because the target application to KSBA moduli spaces is active and the tools are the kind that get reused.","headline":"Extends movable curve classes, slope stability, and a Bogomolov-Gieseker inequality from varieties to smooth proper DM stacks with projective coarse space, as the second paper in a hyperbolicity series.","tokens_in":2133,"tokens_out":401,"would_cite":false,"duration_ms":17714,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Movable curve classes and slope stability extend from varieties to smooth proper DM stacks with projective coarse moduli spaces, yielding a Bogomolov-Gieseker inequality.","keywords":["Deligne-Mumford stacks","movable curve classes","slope stability","Bogomolov-Gieseker inequality","coherent sheaves","moduli spaces","algebraic stacks"],"falsifier":"Exhibiting a coherent sheaf on a smooth proper DM stack with projective coarse moduli space whose Chern classes violate the Bogomolov-Gieseker inequality would falsify the central claim.","tokens_in":2478,"feed_emoji":"📐","tokens_out":593,"duration_ms":18495,"temperature":0.7,"pith_summary":"The paper generalizes theorems on movable curve classes and the slope stability of coherent sheaves, previously known for smooth projective varieties, to the broader setting of smooth proper Deligne-Mumford stacks that have projective coarse moduli spaces. This extension directly produces the Bogomolov-Gieseker inequality for sheaves on the stacks. The move matters because many natural moduli spaces appear as DM stacks rather than varieties, so the new statements open access to stability and positivity results in those cases. The work forms the second part of a series that aims to carry Viehweg hyperbolicity results to this stack setting.","feed_headline":"Slope stability extends from varieties to DM stacks","feed_subtitle":"Generalization produces Bogomolov-Gieseker inequality on smooth proper stacks with projective coarse spaces.","key_machinery":"The extension of the movable cone of curve classes and the associated slope function for coherent sheaves from varieties to DM stacks.","core_discovery":"Movable curve classes and slope stability of coherent sheaves on smooth projective varieties extend to smooth proper DM stacks admitting projective coarse moduli spaces; the resulting slope function then implies the Bogomolov-Gieseker inequality on these stacks.","pith_inferences":["The same extension technique may apply to other classes of algebraic stacks beyond DM stacks.","The inequality could be used to bound the geometry of moduli spaces that arise as coarse spaces of these stacks.","Analogous statements might hold when the coarse space is only quasi-projective rather than projective."],"forward_implications":["The Bogomolov-Gieseker inequality holds for coherent sheaves on all such stacks.","Slope stability with respect to movable classes behaves as it does on varieties.","The inequality supplies a new positivity tool for sheaves on moduli stacks.","The results prepare the ground for hyperbolicity statements on KSBA moduli spaces viewed as stacks."],"fun_headline_variants":["Movable curve classes enable slope stability on DM stacks","Slope stability holds on DM stacks with projective coarse spaces","Generalizing slope stability to smooth proper DM stacks","Movable curves and slope stability extend to DM stacks"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The Deligne-Mumford stacks under consideration are smooth and proper and possess projective coarse moduli spaces.","fun_headline_variants_meta":{"raw":{"variants":["Movable curve classes enable slope stability on DM stacks","Slope stability holds on DM stacks with projective coarse spaces","Generalizing slope stability to smooth proper DM stacks","Movable curves and slope stability extend to DM stacks"]},"model":"grok-4.3","cost_usd":0.006869,"raw_usage":{"total_tokens":3019,"prompt_tokens":489,"num_sources_used":0,"completion_tokens":60,"cost_in_usd_ticks":68690500,"prompt_tokens_details":{"text_tokens":489,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2470,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":489,"tokens_out":60,"duration_ms":19919,"temperature":1.0,"reasoning_tokens":2470,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T20:22:19.277240+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Exhibiting a coherent sheaf on a smooth proper DM stack with projective coarse moduli space whose Chern classes violate the Bogomolov-Gieseker inequality would falsify the central claim.","supporting_citations":[],"review_version":1}