{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:ANLZNH4TU2MSZMXWW2LSJXYIBH","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":"533807ba7e919e37205f73fa4162f613f6a63c0fc5d08c284e6f26a7356681bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-02-11T06:17:41Z","title_canon_sha256":"ec46d2ad3e16f17a689aa9f23773c69fe5df01a38633efaa4c9df75f7e720477"},"schema_version":"1.0","source":{"id":"1302.2402","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.2402","created_at":"2026-05-18T03:34:01Z"},{"alias_kind":"arxiv_version","alias_value":"1302.2402v1","created_at":"2026-05-18T03:34:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.2402","created_at":"2026-05-18T03:34:01Z"},{"alias_kind":"pith_short_12","alias_value":"ANLZNH4TU2MS","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"ANLZNH4TU2MSZMXW","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"ANLZNH4T","created_at":"2026-05-18T12:27:38Z"}],"graph_snapshots":[{"event_id":"sha256:a9fc737b04763c93bc9b9496e390829f74e818918c0f0ecb48a4c5c17710bff5","target":"graph","created_at":"2026-05-18T03:34:01Z","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"},"paper":{"abstract_excerpt":"In a previous paper the authors develop an intersection theory for subspaces of rational functions on an algebraic variety X over complex numbers. In this note, we first extend this intersection theory to an arbitrary algebraically closed ground field. Secondly we give an isomorphism between the group of b-divisors on the birational class of X and the Grothendieck group of the semigroup of subspaces of rational functions on X. The constructed isomorphism moreover preserves the intersection numbers. This provides an alternative approach to b-divisors and their intersection theory.","authors_text":"A. G. Khovanskii, Kiumars Kaveh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-02-11T06:17:41Z","title":"Note on the Grothendieck group of subspaces of rational functions and Shokurov's b-divisors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.2402","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":"sha256:99b1325d0890bdde919ae024230238735c6f31d953ad697c995f419f92d00d0e","target":"record","created_at":"2026-05-18T03:34:01Z","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":"533807ba7e919e37205f73fa4162f613f6a63c0fc5d08c284e6f26a7356681bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-02-11T06:17:41Z","title_canon_sha256":"ec46d2ad3e16f17a689aa9f23773c69fe5df01a38633efaa4c9df75f7e720477"},"schema_version":"1.0","source":{"id":"1302.2402","kind":"arxiv","version":1}},"canonical_sha256":"0357969f93a6992cb2f6b69724df0809cb0ad58620468e9fac0d4ec5946af488","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0357969f93a6992cb2f6b69724df0809cb0ad58620468e9fac0d4ec5946af488","first_computed_at":"2026-05-18T03:34:01.257556Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:34:01.257556Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QQ+sBtbSTc7cdb8VoLlTVXWXZvCF7wOoLSDabibPDE3oYR/OdtG+d17P/MCESrGyoNs+XiO7NsPKTdTNhO9YBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:34:01.258100Z","signed_message":"canonical_sha256_bytes"},"source_id":"1302.2402","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:99b1325d0890bdde919ae024230238735c6f31d953ad697c995f419f92d00d0e","sha256:a9fc737b04763c93bc9b9496e390829f74e818918c0f0ecb48a4c5c17710bff5"],"state_sha256":"be61c7060c558280cbdd18f8d0f7405d45f64605c90e26ec5b54781c9bb7756b"}