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We compute the $A$-valued motivic measure $\\mu_A(\\mathcal{M}(d,\\chi))$ of $\\mathcal{M}(d,\\chi)$ and get explicit formula in codimension $D:=\\rho_d-1$, where $\\rho_d$ is $d-1$ for $d=p$ or $2p$ with $p$ prime, and $7$ otherwise. As a corollary, we get the last $2(D+1)$ Betti numbers of the moduli scheme $M(d,\\chi)$ when $d$ is coprime to $\\chi$."},"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":"1503.06309","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-03-21T15:28:17Z","cross_cats_sorted":[],"title_canon_sha256":"5602937239d4c6a96bb69ed2cae59fa3a5583c5fb77f003f6b8620fd96d95990","abstract_canon_sha256":"e085a25cf190f8f9844778f5fe090369e98a195a322dbb3800fe6ef0feee8665"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:52.169148Z","signature_b64":"ucU+6Zwej91eIs2Z+DTt9QvMfOL7fWujGfE1zPPtP5jfUb6PnUVhdObL1fEVoWyX2rVJMNKZoR6o9BSAaIIsCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7f3e636f5c8ad520c61d8a2da351953c976bab43034a4269f3d865c4ff8bdf3f","last_reissued_at":"2026-05-18T02:19:52.168624Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:52.168624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Motivic measures of the moduli spaces of pure sheaves on $\\mathbb{P}^2$ with all degrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Yao Yuan","submitted_at":"2015-03-21T15:28:17Z","abstract_excerpt":"Let $\\mathcal{M}(d,\\chi)$ be the moduli stack of stable sheaves of rank 0, Euler characteristic $\\chi$ and first Chern class $dH~(d>0)$, with $H$ the hyperplane class in $\\mathbb{P}^2$. We compute the $A$-valued motivic measure $\\mu_A(\\mathcal{M}(d,\\chi))$ of $\\mathcal{M}(d,\\chi)$ and get explicit formula in codimension $D:=\\rho_d-1$, where $\\rho_d$ is $d-1$ for $d=p$ or $2p$ with $p$ prime, and $7$ otherwise. As a corollary, we get the last $2(D+1)$ Betti numbers of the moduli scheme $M(d,\\chi)$ when $d$ is coprime to $\\chi$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.06309","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1503.06309","created_at":"2026-05-18T02:19:52.168715+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.06309v3","created_at":"2026-05-18T02:19:52.168715+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.06309","created_at":"2026-05-18T02:19:52.168715+00:00"},{"alias_kind":"pith_short_12","alias_value":"P47GG324RLKS","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_16","alias_value":"P47GG324RLKSBRQ5","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_8","alias_value":"P47GG324","created_at":"2026-05-18T12:29:34.919912+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/P47GG324RLKSBRQ5RIW2GUMVHS","json":"https://pith.science/pith/P47GG324RLKSBRQ5RIW2GUMVHS.json","graph_json":"https://pith.science/api/pith-number/P47GG324RLKSBRQ5RIW2GUMVHS/graph.json","events_json":"https://pith.science/api/pith-number/P47GG324RLKSBRQ5RIW2GUMVHS/events.json","paper":"https://pith.science/paper/P47GG324"},"agent_actions":{"view_html":"https://pith.science/pith/P47GG324RLKSBRQ5RIW2GUMVHS","download_json":"https://pith.science/pith/P47GG324RLKSBRQ5RIW2GUMVHS.json","view_paper":"https://pith.science/paper/P47GG324","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.06309&json=true","fetch_graph":"https://pith.science/api/pith-number/P47GG324RLKSBRQ5RIW2GUMVHS/graph.json","fetch_events":"https://pith.science/api/pith-number/P47GG324RLKSBRQ5RIW2GUMVHS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P47GG324RLKSBRQ5RIW2GUMVHS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P47GG324RLKSBRQ5RIW2GUMVHS/action/storage_attestation","attest_author":"https://pith.science/pith/P47GG324RLKSBRQ5RIW2GUMVHS/action/author_attestation","sign_citation":"https://pith.science/pith/P47GG324RLKSBRQ5RIW2GUMVHS/action/citation_signature","submit_replication":"https://pith.science/pith/P47GG324RLKSBRQ5RIW2GUMVHS/action/replication_record"}},"created_at":"2026-05-18T02:19:52.168715+00:00","updated_at":"2026-05-18T02:19:52.168715+00:00"}