{"id":"08695c3a-3c47-42ba-9dca-d538bbf39253","arxiv_id":"2606.18357","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Constructs irreducible symplectic varieties of dimension 2n (2≤n≤10) with 16≤b2≤24 as non-trivial terminalisations of finite symplectic quotients of Beauville-Mukai systems on very general K3-del Pezzo double covers.","lead":"This paper constructs irreducible symplectic varieties of even dimension from 4 to 20 whose second Betti number lies between 16 and 24. These examples come from resolving finite quotients of certain systems built on K3 surfaces that double-cover del Pezzo surfaces.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identification of the terminalisation step as weakest matches the only plausible load-bearing point visible from the given abstract. No additional technical flaw (e.g., in Betti-number range, dimension bounds, or generality assumptions) surfaces without the body of the paper. Therefore the provisional UNVERDICTED verdict is left unchanged.","tokens_in":1569,"tokens_out":266,"duration_ms":28437,"concrete_test":"Re-read the full manuscript sections detailing the group actions, quotient singularities, and terminalisation construction (likely §§3-5); confirm that the symplectic form extends and irreducibility is preserved for at least one explicit case with 2≤n≤4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption correctly isolates the central step: that finite symplectic quotients of Beauville-Mukai systems on very general K3-del Pezzo double covers admit non-trivial terminalisations remaining irreducible symplectic. Without the full manuscript, no internal inconsistency, missing hypothesis, or incorrect deduction can be located in the argument itself. The abstract states the construction exists; absent explicit group actions, singularity analysis, or extension of the symplectic form, no load-bearing gap is detectable beyond the provisional status already noted.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript constructs a series of irreducible symplectic varieties of dimension 2n for 2 ≤ n ≤ 10, with second Betti numbers satisfying 16 ≤ b₂ ≤ 24. These varieties are obtained as non-trivial terminalisations of finite symplectic quotients of Beauville-Mukai systems on very general K3-del Pezzo double covers.","tokens_in":1669,"tokens_out":325,"duration_ms":19612,"significance":"If the central construction holds, the work supplies new examples of irreducible symplectic varieties in dimensions up to 20 whose Betti numbers lie in a previously sparsely populated range. The approach via K3-del Pezzo double covers and controlled symplectic quotients offers a systematic geometric source that could be compared with existing deformation classes of hyperkähler varieties and their singular analogues.","major_comments":[],"minor_comments":[{"comment":"The abstract states the dimension range and Betti-number bounds but does not indicate whether the terminalisation process is shown to preserve the symplectic form or the irreducibility of the resulting variety; a dedicated section or proposition establishing these preservation properties would strengthen the exposition.","section":null}],"recommendation":"uncertain","confidential_remarks":"The manuscript appears to be a pure construction paper whose central claim rests on a sequence of geometric steps (quotient, terminalisation, verification of symplectic and irreducibility properties) whose details are not visible in the supplied abstract. Without access to the body of the text, no internal inconsistency can be diagnosed, but the claim cannot be assessed for correctness or novelty relative to the existing literature on irreducible symplectic varieties."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their summary of the manuscript and for noting its potential significance in providing new examples of irreducible symplectic varieties. No specific major comments were provided in the report.","responses":[],"tokens_in":1040,"tokens_out":54,"duration_ms":10871,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is a geometric construction that produces irreducible symplectic varieties for n from 2 to 10, giving dimensions 4 to 20 and second Betti numbers from 16 to 24. These come from non-trivial terminalisations of finite symplectic quotients of Beauville-Mukai systems on very general K3-del Pezzo double covers.\n\nWhat the paper does is combine existing tools in a specific way to fill in some of the thinner parts of the known list. The range is useful because examples with those b2 values are not common, and having explicit families can help test ideas about possible deformation types.\n\nThe soft spot is the central step: showing that the quotients admit terminalisations that stay symplectic and irreducible. The abstract states this works, but the strength depends on the details of the group actions, the singularities, and how the symplectic form extends. If those checks are carried out carefully in the text, the claim holds; without them the argument stays provisional. No sign of circular definitions or parameter fitting.\n\nThis is for people working on hyperkähler geometry and the classification of symplectic varieties. It supplies concrete cases rather than a general theorem.\n\nI would send it for peer review. The construction adds new examples with a stated method, and the gaps are fixable in revision rather than fatal to the approach.","headline":"This paper constructs new irreducible symplectic varieties in dimensions 4-20 with b2 16-24 via K3-del Pezzo double covers, Beauville-Mukai systems, and terminalised quotients.","tokens_in":2174,"tokens_out":358,"would_cite":false,"duration_ms":28781,"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":"Irreducible symplectic varieties of dimensions 4 to 20 arise as terminalisations of finite symplectic quotients of Beauville-Mukai systems on K3-del Pezzo double covers.","keywords":["irreducible symplectic varieties","K3-del Pezzo double covers","Beauville-Mukai systems","symplectic quotients","terminalisations","hyperkahler manifolds","second Betti number","algebraic geometry"],"falsifier":"An explicit computation for any single n between 2 and 10 showing that the terminalisation either fails to be symplectic, introduces non-terminal singularities, or yields a variety whose second Betti number lies outside 16-24.","tokens_in":2459,"feed_emoji":"","tokens_out":708,"duration_ms":23106,"temperature":0.7,"pith_summary":"The paper constructs a family of irreducible symplectic varieties in even dimensions from 4 up to 20. These examples are produced by forming finite symplectic quotients of certain moduli spaces called Beauville-Mukai systems that live on very general double covers of del Pezzo surfaces by K3 surfaces, then taking minimal resolutions of the resulting singularities. The second Betti numbers of the varieties fall in the range 16 to 24. A reader would care because the construction supplies explicit new instances of these higher-dimensional hyperkähler-type objects in a range where concrete examples have been limited.","feed_headline":"New symplectic varieties from K3-del Pezzo quotients","feed_subtitle":"Terminalisations of finite quotients of Beauville-Mukai systems give examples of dimension 2n with b2 between 16 and 24 for n up to 10.","key_machinery":"non-trivial terminalisations of finite symplectic quotients of Beauville-Mukai systems on very general K3-del Pezzo double covers","core_discovery":"We construct a series of irreducible symplectic varieties of dimension 2n, for 2≤n≤10, with second Betti numbers 16≤b2≤24. They arise as non-trivial terminalisations of finite symplectic quotients of Beauville-Mukai systems on very general K3-del Pezzo double covers.","pith_inferences":["These varieties may supply test cases for conjectures on the possible values of b2 for irreducible symplectic varieties in each dimension.","Further Hodge numbers or the Fujiki relation constant could be computed directly from the construction for small n.","The same quotient-and-terminalise method might apply to other families of surfaces carrying Beauville-Mukai systems.","The resulting moduli spaces of these new varieties could be compared with known deformation types."],"forward_implications":["Examples exist in every even dimension from 4 through 20.","The second Betti numbers of these varieties lie between 16 and 24 inclusive.","The source objects are Beauville-Mukai systems on very general K3 surfaces that double-cover del Pezzo surfaces.","The group actions used to form the quotients are finite and symplectic.","The terminalisations are non-trivial, meaning they differ from the original quotients."],"fun_headline_variants":["Irreducible symplectic varieties via K3-del Pezzo double covers","Symplectic varieties from K3-del Pezzo quotients","Symplectic varieties via terminalised K3-del Pezzo quotients","Dim 2n symplectic varieties with b2 16-24 from K3-del Pezzo"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The finite symplectic quotients of the Beauville-Mukai systems on very general K3-del Pezzo double covers admit non-trivial terminalisations that remain irreducible symplectic varieties.","fun_headline_variants_meta":{"raw":{"variants":["Irreducible symplectic varieties via K3-del Pezzo double covers","Symplectic varieties from K3-del Pezzo quotients","Symplectic varieties via terminalised K3-del Pezzo quotients","Dim 2n symplectic varieties with b2 16-24 from K3-del Pezzo"]},"model":"grok-4.3","cost_usd":0.013249,"raw_usage":{"total_tokens":5653,"prompt_tokens":492,"num_sources_used":0,"completion_tokens":77,"cost_in_usd_ticks":132487000,"prompt_tokens_details":{"text_tokens":492,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":5084,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":492,"tokens_out":77,"duration_ms":49854,"temperature":1.0,"reasoning_tokens":5084,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T22:20:11.333824+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit computation for any single n between 2 and 10 showing that the terminalisation either fails to be symplectic, introduces non-terminal singularities, or yields a variety whose second Betti number lies outside 16-24.","supporting_citations":[],"review_version":1}