{"id":"6cba7eda-7324-4e37-b4f4-5718290c22ab","arxiv_id":"2606.29803","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":3.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Finite flat quotients of normal projective varieties have numerical groups and positive cones related to those of the original variety.","lead":"The paper studies positivity properties of finite flat quotients of normal projective varieties and relates their numerical groups and positive cones to those of the original variety. Smart generalists might read it to understand how quotients transform positivity structures in algebraic geometry.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's assessment is limited to the abstract and correctly flags low confidence. With no full-text details supplied that would reveal a weak link in the argument, no load-bearing concern can be identified. The central claim is a standard type of relation under group quotients and does not, on its face, contradict known results in algebraic geometry.","tokens_in":1491,"tokens_out":275,"duration_ms":22912,"concrete_test":"Extract the precise statement of the main theorem relating the cones (e.g., Nef or Mori cones) and check whether the proof invokes only the finite flat hypothesis plus normality/projectivity, or whether an unstated condition (e.g., Q-factoriality or freeness of the action) is required; if the theorem follows exactly from the stated hypotheses, the claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract states that numerical groups and positive cones of finite flat quotients of a normal projective variety are related to those of the original variety. The setting (finite flat quotients, normal projective source) is the standard one in which such descent or pushforward relations for numerical classes and cones are expected to hold via the quotient map. No internal inconsistency, hidden assumption, or unsupported step is detectable from the provided information.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript studies positivity properties of finite flat quotients of a normal projective variety, claiming that the numerical groups and positive cones of the quotient varieties are related to those of the original variety.","tokens_in":1534,"tokens_out":227,"duration_ms":24561,"significance":"The topic addresses a standard question in algebraic geometry concerning descent of numerical classes and positivity under finite flat morphisms. If the relations were stated precisely with proofs, the work could supply useful comparison maps between N^1 or N_1 groups and their cones. However, the provided text contains only the abstract and no derivations, theorems, or examples, so significance cannot be evaluated.","major_comments":[{"comment":"No theorems, propositions, or proofs are present in the manuscript. The central claim that numerical groups and positive cones are related cannot be checked for correctness or even stated precisely, rendering the paper unverifiable.","section":null}],"minor_comments":[],"recommendation":"reject","confidential_remarks":"The submission appears incomplete; only the abstract is supplied despite the prompt indicating full text availability."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their report. We acknowledge that the version under review contains only the abstract and no theorems or proofs, which prevents verification of the claims.","responses":[{"response":"The referee is correct: the submitted manuscript consists solely of the abstract and provides no derivations, theorems, or examples. Without these, the precise statements relating N^1, N_1, and the positive cones under finite flat quotients cannot be evaluated. We will prepare a revised version that includes the full statements, proofs, and any necessary examples.","revision_made":"yes","referee_comment":"No theorems, propositions, or proofs are present in the manuscript. The central claim that numerical groups and positive cones are related cannot be checked for correctness or even stated precisely, rendering the paper unverifiable."}],"tokens_in":992,"tokens_out":216,"duration_ms":22593,"standing_objections":["The current manuscript text contains no theorems or proofs, so the central claims remain unverifiable until a complete version is supplied."]},"desk_editor":{"model":"grok-4.3","letter":"The core claim is that numerical groups and positive cones on finite flat quotients of a normal projective variety relate back to those on the original variety. This is the one thing to take away.\n\nThe work studies positivity properties in this quotient setting. It may give explicit relations via the quotient map, which can be convenient when handling group actions or symmetric cases.\n\nNothing in the abstract points to new tools, first-principles derivations, or broad applications. The topic is a standard one in algebraic geometry, so the result looks like a routine extension rather than a shift in approach.\n\nThe clearest limitation is that only the abstract is visible here. Without proofs, examples, or explicit statements of the relations, it is impossible to judge whether the arguments are solid or whether they add anything beyond what follows from pushforward of classes under finite flat maps. The setting itself is the usual one where such relations are expected, so no internal contradiction shows up.\n\nThis is for people already working on positivity cones and numerical groups in algebraic geometry. A specialist might pull a reference or calculation from it if the full proofs are clean, but a general reader will not find much.\n\nIt is worth sending to a referee to check the details and see whether the relations are stated and proved in a useful way.","headline":"The paper relates numerical groups and positivity cones between a normal projective variety and its finite flat quotients, but the abstract gives no sign of new mechanisms or detailed proofs.","tokens_in":2030,"tokens_out":341,"would_cite":false,"duration_ms":45539,"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":"Finite flat quotients of a normal projective variety have numerical groups and positive cones related to those of the original variety.","keywords":["positive cones","finite quotients","normal projective varieties","numerical groups","positivity properties","finite flat morphisms"],"falsifier":"An explicit finite flat quotient in which the positive cone of the quotient fails to match the image of the original positive cone under the induced map on numerical classes.","tokens_in":2384,"feed_emoji":"📐","tokens_out":337,"duration_ms":23711,"temperature":0.7,"pith_summary":"The paper examines how positivity properties behave under finite flat quotients of a normal projective variety. It establishes explicit relations between the numerical groups of the quotient and those of the original variety, and likewise for their positive cones. A sympathetic reader cares because these relations let one transfer information about numerical equivalence and positivity from the base variety to its quotients without separate computation. The setting is restricted to finite flat morphisms from normal projective varieties.","feed_headline":"Finite quotients link positive cones to original variety","feed_subtitle":"Numerical groups and positivity structures of quotients connect to those of the base normal projective variety via the quotient map.","key_machinery":"The finite flat quotient morphism, which induces corresponding maps between numerical groups and between positive cones.","core_discovery":"The numerical groups and the positive cones of these quotient varieties are related to those of the original variety.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Finite quotients connect positive cones to originals","Positive cones of quotients relate to original variety","Quotient varieties share positive cones with originals","Normal quotients link cones to base variety","Finite quotients tie numerical groups to original cones"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The quotients are finite and flat, and the original variety is normal and projective.","fun_headline_variants_meta":{"raw":{"variants":["Finite quotients connect positive cones to originals","Positive cones of quotients relate to original variety","Quotient varieties share positive cones with originals","Normal quotients link cones to base variety","Finite quotients tie numerical groups to original cones"]},"model":"grok-4.3","cost_usd":0.008929,"raw_usage":{"total_tokens":3879,"prompt_tokens":400,"num_sources_used":0,"completion_tokens":67,"cost_in_usd_ticks":89287000,"prompt_tokens_details":{"text_tokens":400,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3412,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":400,"tokens_out":67,"duration_ms":40147,"temperature":1.0,"reasoning_tokens":3412,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T04:36:39.452286+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit finite flat quotient in which the positive cone of the quotient fails to match the image of the original positive cone under the induced map on numerical classes.","supporting_citations":[],"review_version":1}