{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:BDQYPEHOEUAUZMWPPE4PNW3TFU","short_pith_number":"pith:BDQYPEHO","schema_version":"1.0","canonical_sha256":"08e18790ee25014cb2cf7938f6db732d166e4fc780a67027ce4dd5c146d3c54f","source":{"kind":"arxiv","id":"2307.02822","version":1},"attestation_state":"computed","paper":{"title":"A note on stable toric sheaves of low rank","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Carl Tipler","submitted_at":"2023-07-06T07:42:43Z","abstract_excerpt":"Kaneyama and Klyachko have shown that any torus equivariant vector bundle of rank $r$ over $\\mathbb{CP}^n$ splits if $r < n$. In particular, any such bundle is not slope stable. In contrast, we provide explicit examples of stable equivariant reflexive sheaves of rank $r$ on any polarised toric variety $(X, L)$, for $2 \\leq r < \\mathrm{dim}(X) + \\mathrm{rank}(\\mathrm{Pic}(X))$, and show that the dimension of their singular locus is strictly bounded by $n - r$."},"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":"2307.02822","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2023-07-06T07:42:43Z","cross_cats_sorted":[],"title_canon_sha256":"094d432e8da1851c2a904636d75f4693cf9f0f2b9a168fb77cbbc5ad7db0db18","abstract_canon_sha256":"945599f7731d78b4dd31dec38eeca6a89612c68e87c7d24ae45eabe355fff979"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:28:24.865229Z","signature_b64":"lASiRQNw0dSpKmgypD6JNnvaB+hzo+ZUgPQep9udz7wPlZuEBa7hQLNqATmQ9R2gP8c9JArumGZAyJqgLh4+AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"08e18790ee25014cb2cf7938f6db732d166e4fc780a67027ce4dd5c146d3c54f","last_reissued_at":"2026-07-05T06:28:24.864773Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:28:24.864773Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on stable toric sheaves of low rank","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Carl Tipler","submitted_at":"2023-07-06T07:42:43Z","abstract_excerpt":"Kaneyama and Klyachko have shown that any torus equivariant vector bundle of rank $r$ over $\\mathbb{CP}^n$ splits if $r < n$. In particular, any such bundle is not slope stable. In contrast, we provide explicit examples of stable equivariant reflexive sheaves of rank $r$ on any polarised toric variety $(X, L)$, for $2 \\leq r < \\mathrm{dim}(X) + \\mathrm{rank}(\\mathrm{Pic}(X))$, and show that the dimension of their singular locus is strictly bounded by $n - r$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.02822","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2307.02822/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2307.02822","created_at":"2026-07-05T06:28:24.864832+00:00"},{"alias_kind":"arxiv_version","alias_value":"2307.02822v1","created_at":"2026-07-05T06:28:24.864832+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2307.02822","created_at":"2026-07-05T06:28:24.864832+00:00"},{"alias_kind":"pith_short_12","alias_value":"BDQYPEHOEUAU","created_at":"2026-07-05T06:28:24.864832+00:00"},{"alias_kind":"pith_short_16","alias_value":"BDQYPEHOEUAUZMWP","created_at":"2026-07-05T06:28:24.864832+00:00"},{"alias_kind":"pith_short_8","alias_value":"BDQYPEHO","created_at":"2026-07-05T06:28:24.864832+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.23842","citing_title":"Polynomial stability conditions for vector bundles: Positivity, equivariance and blow-ups","ref_index":43,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BDQYPEHOEUAUZMWPPE4PNW3TFU","json":"https://pith.science/pith/BDQYPEHOEUAUZMWPPE4PNW3TFU.json","graph_json":"https://pith.science/api/pith-number/BDQYPEHOEUAUZMWPPE4PNW3TFU/graph.json","events_json":"https://pith.science/api/pith-number/BDQYPEHOEUAUZMWPPE4PNW3TFU/events.json","paper":"https://pith.science/paper/BDQYPEHO"},"agent_actions":{"view_html":"https://pith.science/pith/BDQYPEHOEUAUZMWPPE4PNW3TFU","download_json":"https://pith.science/pith/BDQYPEHOEUAUZMWPPE4PNW3TFU.json","view_paper":"https://pith.science/paper/BDQYPEHO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2307.02822&json=true","fetch_graph":"https://pith.science/api/pith-number/BDQYPEHOEUAUZMWPPE4PNW3TFU/graph.json","fetch_events":"https://pith.science/api/pith-number/BDQYPEHOEUAUZMWPPE4PNW3TFU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BDQYPEHOEUAUZMWPPE4PNW3TFU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BDQYPEHOEUAUZMWPPE4PNW3TFU/action/storage_attestation","attest_author":"https://pith.science/pith/BDQYPEHOEUAUZMWPPE4PNW3TFU/action/author_attestation","sign_citation":"https://pith.science/pith/BDQYPEHOEUAUZMWPPE4PNW3TFU/action/citation_signature","submit_replication":"https://pith.science/pith/BDQYPEHOEUAUZMWPPE4PNW3TFU/action/replication_record"}},"created_at":"2026-07-05T06:28:24.864832+00:00","updated_at":"2026-07-05T06:28:24.864832+00:00"}