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We show that the divisibility property of the Poincar\\'e polynomial of $M_{\\beta}$, proposed by Choi-van Garrel-Katz-Takahashi follows from Bousseau's conjectural refined sheaves/Gromov-Witten correspondence. Since this correspondence is known for $S=\\mathbb{P}^2$, our result proves Choi-van Garrel-Katz-Takahashi's conjecture in this case.\n  For $S=\\mathbb{P}^2$, our proof also introduces a novel approach to computing the Poin"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2501.05622","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2025-01-09T23:42:10Z","cross_cats_sorted":[],"title_canon_sha256":"d95a0267b95dedc6a0f4875873321f8fcf0a90e3a6d9ec3fd984c6005acdc649","abstract_canon_sha256":"42325e6eeede924668fc350d8d9ae49418b4e21a20926d0b0a1bf4b20fb07aff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:27:23.350631Z","signature_b64":"ZuYIRCBbBT7uS4Rjc6nfwQHRdxG+c69VCMXhs8rS8dG3b/Yq+479UQNBx1UIsowG29NKokoZW7x6RpuTvPm9DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d90f7df4b50e71f9961581c61cb53f174b0e222c9149b420bd34eaaeb1ab22ee","last_reissued_at":"2026-07-05T10:27:23.349819Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:27:23.349819Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Poincar\\'e polynomials of moduli spaces of one-dimensional sheaves on the projective plane","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Longting Wu, Shuai Guo, with an appendix by Miguel Moreira","submitted_at":"2025-01-09T23:42:10Z","abstract_excerpt":"Let $M_{\\beta}$ denote the moduli space of stable one-dimensional sheaves on a del Pezzo surface $S$, supported on curves of class $\\beta$ with Euler characteristic one. 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