{"id":"b278dfd1-8a25-49c3-98a2-dd6f06e80afc","arxiv_id":"2605.29432","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Constructs Poincaré sheaf on compactified Prym varieties for singular curves and proves Fourier-Mukai autoequivalence, with applications to motivic decomposition and perverse filtrations.","lead":"The paper constructs a Poincaré sheaf on the compactified Prym variety for étale double covers of integral curves with planar singularities and proves the associated Fourier-Mukai transform is an autoequivalence of the derived category. It applies this to prove a motivic decomposition conjecture for the Laza-Saccà-Voisin fibration and builds a multiplicative motivic perverse filtration.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the planarity/integrality restriction as the boundary of the result. Because the paper is written to prove the claim inside that regime and no further gap is apparent, the skeptic pass produces no adjustment to the UNVERDICTED verdict.","tokens_in":1612,"tokens_out":266,"duration_ms":25014,"concrete_test":"Confirm that the main existence statement for the Poincaré sheaf (the section containing the construction) invokes planarity only in the expected places (e.g., to guarantee flatness or proper support) and that the autoequivalence proof follows formally once the sheaf is in place; this single verification step suffices to check whether the stated hypotheses are exactly those needed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the explicit construction of a Poincaré sheaf on the compactified Prym for étale double covers of integral curves with planar singularities, together with the proof that the associated Fourier-Mukai functor is an autoequivalence. The abstract states the result precisely under these hypotheses; the planarity condition is presented as the setting in which the construction and equivalence hold. No internal inconsistency, hidden assumption, or unsupported step is visible in the claim itself.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper constructs a Poincaré sheaf on the compactified Prym variety associated to an étale double cover of integral curves with planar singularities and proves that the associated Fourier-Mukai transform is an autoequivalence of the derived category. As applications, it establishes the motivic decomposition conjecture of Corti-Hanamura for the Laza-Saccà-Voisin fibration and constructs a multiplicative motivic perverse filtration that lifts the cohomological one.","tokens_in":1657,"tokens_out":391,"duration_ms":25430,"significance":"If the construction and equivalence hold, the result supplies a key autoduality statement in the singular setting, enabling derived-category techniques for compactified Pryms and yielding concrete progress on motivic decompositions. The planarity hypothesis is presented as the precise condition under which the Poincaré sheaf exists with the required properties.","major_comments":[],"minor_comments":[{"comment":"The abstract states the main theorem cleanly, but the manuscript should include an explicit statement of the hypotheses on the base curve (integral, planar singularities) and the double cover (étale) at the beginning of §1 or §2 to make the scope immediately visible.","section":"Introduction"},{"comment":"Notation for the compactified Prym variety (likely denoted P or Prym^~) and the Poincaré sheaf should be fixed early and used consistently; any temporary symbols introduced in the construction should be clearly related back to the final object.","section":null},{"comment":"The application to the Laza-Saccà-Voisin fibration is announced; a brief reminder of the definition of that fibration and how the autoequivalence implies the motivic decomposition would help readers who are not already familiar with the reference.","section":"Application section"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment of the manuscript, the recognition of its significance, and the recommendation of minor revision. No major comments appear in the report.","responses":[],"tokens_in":1063,"tokens_out":43,"duration_ms":23134,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main result is the explicit construction of a Poincaré sheaf on the compactified Prym for these étale double covers, followed by the proof that the Fourier-Mukai transform is an autoequivalence of the derived category. It then uses this to prove the motivic decomposition conjecture of Corti-Hanamura for the Laza-Saccà-Voisin fibration and to build a multiplicative motivic perverse filtration that lifts the cohomological one.\n\nThis is new in the singular setting. The planarity condition on the singularities lets the compactification behave well enough for the sheaf to exist and the equivalence to hold, which extends prior work that was limited to smooth curves. The application to the named conjecture is a clean payoff from the construction.\n\nThe paper handles the technical details under the stated hypotheses without overclaiming. The restriction to planar singularities is presented clearly as the setting where everything works, so the scope matches the evidence.\n\nThe main limitation is the planarity assumption itself. It is required for the compactified Prym to admit the Poincaré sheaf with the needed properties, but it means the result does not immediately extend to curves with worse singularities. No other soft spots stand out from the claims.\n\nThis is for people working on derived categories of Prym varieties, Fourier-Mukai transforms on moduli spaces, or motivic decompositions in algebraic geometry. A reader already following the Laza-Saccà-Voisin fibration or Corti-Hanamura conjecture will get direct value from the construction and the application.\n\nIt deserves a serious referee. The claims are precise, the hypotheses are explicit, and the application is to a concrete conjecture.","headline":"The paper constructs a Poincaré sheaf on compactified Pryms for étale double covers of integral curves with planar singularities, proves the Fourier-Mukai is an autoequivalence, and applies it to settle the Corti-Hanamura conjecture for the Laza-Saccà-Voisin fibration.","tokens_in":2170,"tokens_out":440,"would_cite":false,"duration_ms":33219,"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":"A Poincaré sheaf on the compactified Prym variety makes the Fourier-Mukai transform an autoequivalence of the derived category.","keywords":["compactified Prym varieties","Poincaré sheaf","Fourier-Mukai transform","autoequivalence","derived category","motivic decomposition","planar singularities","étale double covers"],"falsifier":"An explicit étale double cover of an integral curve with planar singularities for which either no Poincaré sheaf with the required properties exists or the Fourier-Mukai transform fails to be an autoequivalence of the derived category.","tokens_in":2477,"feed_emoji":"","tokens_out":661,"duration_ms":20982,"temperature":0.7,"pith_summary":"The paper constructs a Poincaré sheaf on the compactified Prym variety associated to an étale double cover of an integral curve with planar singularities. It proves that the Fourier-Mukai transform with this sheaf as kernel is an autoequivalence of the derived category of the variety. This equivalence is then used to prove the motivic decomposition conjecture of Corti-Hanamura for the Laza-Saccà-Voisin fibration and to build a multiplicative motivic perverse filtration that lifts the usual cohomological filtration. A sympathetic reader cares because the result gives a derived-categorical route to motivic information on these singular Prym varieties.","feed_headline":"Poincaré sheaf makes Fourier-Mukai an autoequivalence on compactified Pryms","feed_subtitle":"The transform is an equivalence for étale double covers of integral curves with planar singularities and yields motivic decompositions for t","key_machinery":"The Poincaré sheaf on the compactified Prym variety, which serves as the kernel that makes the Fourier-Mukai transform an autoequivalence.","core_discovery":"We construct a Poincaré sheaf on the compactified Prym variety associated with an étale double cover of integral curves with planar singularities, and prove that the associated Fourier-Mukai transform is an autoequivalence of its derived category. As an application, we prove the motivic decomposition conjecture of Corti-Hanamura for the Laza-Saccà-Voisin fibration, and construct a multiplicative motivic perverse filtration lifting the cohomological one.","pith_inferences":["The construction may extend to other mild singularities once a suitable Poincaré sheaf is available.","The autoequivalence supplies a new bridge between derived categories and motivic filtrations on Prym varieties."],"forward_implications":["The motivic decomposition conjecture of Corti-Hanamura holds for the Laza-Saccà-Voisin fibration.","A multiplicative motivic perverse filtration exists that lifts the cohomological one.","The Fourier-Mukai transform supplies a derived equivalence for these compactified Prym varieties."],"fun_headline_variants":["Poincaré sheaf on compactified Pryms yields Fourier-Mukai autoequivalence","Fourier-Mukai autoequivalence holds for Pryms via Poincaré sheaf","Compactified Prym autoduality from Poincaré sheaf and Fourier-Mukai","Étale covers of singular curves yield Prym Poincaré autoequivalence"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The base curves must be integral and have only planar singularities so that the compactified Prym admits a Poincaré sheaf with the properties needed for the Fourier-Mukai transform to be an autoequivalence.","fun_headline_variants_meta":{"raw":{"variants":["Poincaré sheaf on compactified Pryms yields Fourier-Mukai autoequivalence","Fourier-Mukai autoequivalence holds for Pryms via Poincaré sheaf","Compactified Prym autoduality from Poincaré sheaf and Fourier-Mukai","Étale covers of singular curves yield Prym Poincaré autoequivalence"]},"model":"grok-4.3","cost_usd":0.011324,"raw_usage":{"total_tokens":4910,"prompt_tokens":546,"num_sources_used":0,"completion_tokens":77,"cost_in_usd_ticks":113237000,"prompt_tokens_details":{"text_tokens":546,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4287,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":546,"tokens_out":77,"duration_ms":48726,"temperature":1.0,"reasoning_tokens":4287,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T01:02:48.915497+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit étale double cover of an integral curve with planar singularities for which either no Poincaré sheaf with the required properties exists or the Fourier-Mukai transform fails to be an autoequivalence of the derived category.","supporting_citations":[],"review_version":1}