{"paper":{"title":"Tilt-stability on singular schemes and Bogomolov-Gieseker-type inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Tilt-stability extends to singular schemes and supports a generalized Bogomolov-Gieseker inequality conjecture for singular threefolds.","cross_cats":[],"primary_cat":"math.AG","authors_text":"Tianle Mao, Zhiyu Liu","submitted_at":"2026-05-13T17:36:44Z","abstract_excerpt":"We generalize the framework of tilt-stability to singular schemes and formulate the generalized Bogomolov-Gieseker inequality conjecture of Bayer-Macr\\`i-Toda for singular threefolds. We also develop relative versions of these constructions, generalizing corresponding results in [BLM+21]. Along the way, we establish Bogomolov-Gieseker-type inequalities for semistable sheaves on any projective scheme.\n  By extending previous techniques, we verify the conjecture for all Fano threefolds with canonical Gorenstein $\\mathbb{Q}$-factorial singularities and a series of singular Calabi-Yau threefolds. "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We generalize the framework of tilt-stability to singular schemes and formulate the generalized Bogomolov-Gieseker inequality conjecture of Bayer-Macrì-Toda for singular threefolds.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The schemes are projective and the singularities satisfy conditions such as canonical Gorenstein Q-factorial for the verifications to apply.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Tilt-stability is extended to singular schemes, a generalized Bogomolov-Gieseker conjecture is formulated and verified for certain singular threefolds, and stability conditions are constructed on relative Kuznetsov components.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Tilt-stability extends to singular schemes and supports a generalized Bogomolov-Gieseker inequality conjecture for singular threefolds.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c3c904e316cd2e0931c38314bc0c81b9705b225e9184b6d26635816595435170"},"source":{"id":"2605.13808","kind":"arxiv","version":1},"verdict":{"id":"63f2851b-126e-4e0a-8595-b4d7d80af1d4","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:36:15.522358Z","strongest_claim":"We generalize the framework of tilt-stability to singular schemes and formulate the generalized Bogomolov-Gieseker inequality conjecture of Bayer-Macrì-Toda for singular threefolds.","one_line_summary":"Tilt-stability is extended to singular schemes, a generalized Bogomolov-Gieseker conjecture is formulated and verified for certain singular threefolds, and stability conditions are constructed on relative Kuznetsov components.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The schemes are projective and the singularities satisfy conditions such as canonical Gorenstein Q-factorial for the verifications to apply.","pith_extraction_headline":"Tilt-stability extends to singular schemes and supports a generalized Bogomolov-Gieseker inequality conjecture for singular threefolds."},"references":{"count":16,"sample":[{"doi":"","year":2015,"title":"Wall-crossing implies Brill–Noether: applications of stability conditions on surfaces","work_id":"eb91eb73-0e7d-42d3-b4ff-c72ef86117d3","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Sur le produit de vari\\'et\\'es localement factorielles ou Q-factorielles","work_id":"6f36b3d8-5390-4efc-9ef7-88511af38c91","ref_index":2,"cited_arxiv_id":"1104.1861","is_internal_anchor":true},{"doi":"","year":null,"title":"Mukai bundles on Fano three- folds.arXiv preprint, arXiv:2402.07154,","work_id":"e0137b37-8def-4c9e-acd2-f58e6ae3311c","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Mukai models of Fano varieties","work_id":"8c5c6cf8-90df-4c44-b356-6fff4cd04c61","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"EMS Press, Berlin, [2023]©2023","work_id":"a300b55d-3834-49b5-bf44-14a1ccf3a4f2","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":16,"snapshot_sha256":"4d6ffe2d423aaed40ba0e43c23c6f168398e052bec3944c1c48c086da146de7d","internal_anchors":2},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}