{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:2B6DUEXC7JR2RX3GUIIRU6TP5X","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"fad163d1a394a212feb20c61d6d51c289059d6bff5307b34c70846968447934b","cross_cats_sorted":[],"license":"","primary_cat":"math.AG","submitted_at":"2004-06-19T01:15:24Z","title_canon_sha256":"8d33c2592084c91cb25d9926c01198943188061e9299e292fc1bc97f0363128a"},"schema_version":"1.0","source":{"id":"math/0406384","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0406384","created_at":"2026-07-04T14:38:43Z"},{"alias_kind":"arxiv_version","alias_value":"math/0406384v1","created_at":"2026-07-04T14:38:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0406384","created_at":"2026-07-04T14:38:43Z"},{"alias_kind":"pith_short_12","alias_value":"2B6DUEXC7JR2","created_at":"2026-07-04T14:38:43Z"},{"alias_kind":"pith_short_16","alias_value":"2B6DUEXC7JR2RX3G","created_at":"2026-07-04T14:38:43Z"},{"alias_kind":"pith_short_8","alias_value":"2B6DUEXC","created_at":"2026-07-04T14:38:43Z"}],"graph_snapshots":[{"event_id":"sha256:34d71f79ea1b8d46710c8f0a1369762fac8172732c3d4f8f54f668e2f66d1db5","target":"graph","created_at":"2026-07-04T14:38:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0406384/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We define the affine stratification number asn X of a scheme X. For X equidimensional, it is the minimal number k such that there is a stratification of X by locally closed affine subschemes of codimension at most k. We show that the affine stratification number is well-behaved, and bounds many aspects of the topological complexity of the scheme, such as vanishing of cohomology groups of quasicoherent, constructible, and l-adic sheaves. We explain how to bound asn X in practice. We give a series of conjectures (the first by E. Looijenga) bounding the affine stratification number of various mod","authors_text":"Mike Roth, Ravi Vakil","cross_cats":[],"headline":"","license":"","primary_cat":"math.AG","submitted_at":"2004-06-19T01:15:24Z","title":"The affine stratification number and the moduli space of curves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0406384","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:43817e1123ca1ee85f52310a4a1a5cac6859cc84b1b03baf86996eba9cd01635","target":"record","created_at":"2026-07-04T14:38:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"fad163d1a394a212feb20c61d6d51c289059d6bff5307b34c70846968447934b","cross_cats_sorted":[],"license":"","primary_cat":"math.AG","submitted_at":"2004-06-19T01:15:24Z","title_canon_sha256":"8d33c2592084c91cb25d9926c01198943188061e9299e292fc1bc97f0363128a"},"schema_version":"1.0","source":{"id":"math/0406384","kind":"arxiv","version":1}},"canonical_sha256":"d07c3a12e2fa63a8df66a2111a7a6fedd5a3e44fac8fe99914bc08f9fe2f6ec2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d07c3a12e2fa63a8df66a2111a7a6fedd5a3e44fac8fe99914bc08f9fe2f6ec2","first_computed_at":"2026-07-04T14:38:43.891230Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:38:43.891230Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jfhGKnMtnaaf+nSw3tJ1GyaCHGbQpYNi5P3STJV9B7pTA8dzJGowzuwbSxnUdw4cb95qe62U8iCyV4QTmbKqCA==","signature_status":"signed_v1","signed_at":"2026-07-04T14:38:43.891626Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0406384","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:43817e1123ca1ee85f52310a4a1a5cac6859cc84b1b03baf86996eba9cd01635","sha256:34d71f79ea1b8d46710c8f0a1369762fac8172732c3d4f8f54f668e2f66d1db5"],"state_sha256":"53481e51f46b1ee622ea7127721a05bd3272d7db1514099e68caa45aee07330e"}