{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:VNZ5MAOR4OBCIGP3W7ESNGYS6W","short_pith_number":"pith:VNZ5MAOR","canonical_record":{"source":{"id":"1904.01077","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-04-01T19:31:15Z","cross_cats_sorted":[],"title_canon_sha256":"28cfde70fd8b254fd882b52756e3754728b7523d5a5867a5f21102ff3b019773","abstract_canon_sha256":"24a52a386e6a2628025e6d13bcce376b45ec2cc83d6585cdcc79c68ae14f0e66"},"schema_version":"1.0"},"canonical_sha256":"ab73d601d1e3822419fbb7c9269b12f5a6a81f7e0ba0a91e057eaf27dee8f8c7","source":{"kind":"arxiv","id":"1904.01077","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.01077","created_at":"2026-05-17T23:39:24Z"},{"alias_kind":"arxiv_version","alias_value":"1904.01077v2","created_at":"2026-05-17T23:39:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.01077","created_at":"2026-05-17T23:39:24Z"},{"alias_kind":"pith_short_12","alias_value":"VNZ5MAOR4OBC","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"VNZ5MAOR4OBCIGP3","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"VNZ5MAOR","created_at":"2026-05-18T12:33:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:VNZ5MAOR4OBCIGP3W7ESNGYS6W","target":"record","payload":{"canonical_record":{"source":{"id":"1904.01077","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-04-01T19:31:15Z","cross_cats_sorted":[],"title_canon_sha256":"28cfde70fd8b254fd882b52756e3754728b7523d5a5867a5f21102ff3b019773","abstract_canon_sha256":"24a52a386e6a2628025e6d13bcce376b45ec2cc83d6585cdcc79c68ae14f0e66"},"schema_version":"1.0"},"canonical_sha256":"ab73d601d1e3822419fbb7c9269b12f5a6a81f7e0ba0a91e057eaf27dee8f8c7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:24.497868Z","signature_b64":"fjpv3ksqWRUby+WMPaWN44e4Sf9qN0cfpQL7uAA6truyWSgZTtGck19AStCeHMdHbiT+CK4txdsI6tDcq4JCDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab73d601d1e3822419fbb7c9269b12f5a6a81f7e0ba0a91e057eaf27dee8f8c7","last_reissued_at":"2026-05-17T23:39:24.495615Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:24.495615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1904.01077","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:39:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XeDEs9yQw+J910DVQ9t2/LTqqinoMmGs9xz/3XyOVcY+kHkLxf7Q9qsGBq4baSYMj2RWgA3Azms9xS0kI38pAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T02:43:33.961053Z"},"content_sha256":"9974d971c64491e2817f9f4418d9ef0e5b1b5d8c6dbd4881308cedb2cc1aa067","schema_version":"1.0","event_id":"sha256:9974d971c64491e2817f9f4418d9ef0e5b1b5d8c6dbd4881308cedb2cc1aa067"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:VNZ5MAOR4OBCIGP3W7ESNGYS6W","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"From Cracked Polytopes to Fano Threefolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Thomas Prince","submitted_at":"2019-04-01T19:31:15Z","abstract_excerpt":"We construct Fano threefolds with very ample anti-canonical bundle and Picard rank greater than one from cracked polytopes - polytopes whose intersection with a complete fan forms a set of unimodular polytopes - using Laurent inversion; a method developed jointly with Coates-Kasprzyk. We also give constructions of rank one Fano threefolds from cracked polytopes, following work of Christophersen-Ilten and Galkin. We explore the problem of classifying polytopes cracked along a given fan in three dimensions, and classify the unimodular polytopes which can occur as 'pieces' of a cracked polytope."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.01077","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:39:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5vIB85tlomb3Bax9TuLQNjn+YQU6CsDyUnpsMtW5gn67PxaXdfKaDxj6ctoJ0FuZp+uiqGUXZWn+TmztnPePCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T02:43:33.961403Z"},"content_sha256":"7a26f1933a7dff446823ba313e5031f267a5cb61033b2bef2e5b22fa58777465","schema_version":"1.0","event_id":"sha256:7a26f1933a7dff446823ba313e5031f267a5cb61033b2bef2e5b22fa58777465"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VNZ5MAOR4OBCIGP3W7ESNGYS6W/bundle.json","state_url":"https://pith.science/pith/VNZ5MAOR4OBCIGP3W7ESNGYS6W/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VNZ5MAOR4OBCIGP3W7ESNGYS6W/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T02:43:33Z","links":{"resolver":"https://pith.science/pith/VNZ5MAOR4OBCIGP3W7ESNGYS6W","bundle":"https://pith.science/pith/VNZ5MAOR4OBCIGP3W7ESNGYS6W/bundle.json","state":"https://pith.science/pith/VNZ5MAOR4OBCIGP3W7ESNGYS6W/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VNZ5MAOR4OBCIGP3W7ESNGYS6W/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:VNZ5MAOR4OBCIGP3W7ESNGYS6W","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":"24a52a386e6a2628025e6d13bcce376b45ec2cc83d6585cdcc79c68ae14f0e66","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-04-01T19:31:15Z","title_canon_sha256":"28cfde70fd8b254fd882b52756e3754728b7523d5a5867a5f21102ff3b019773"},"schema_version":"1.0","source":{"id":"1904.01077","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.01077","created_at":"2026-05-17T23:39:24Z"},{"alias_kind":"arxiv_version","alias_value":"1904.01077v2","created_at":"2026-05-17T23:39:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.01077","created_at":"2026-05-17T23:39:24Z"},{"alias_kind":"pith_short_12","alias_value":"VNZ5MAOR4OBC","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"VNZ5MAOR4OBCIGP3","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"VNZ5MAOR","created_at":"2026-05-18T12:33:30Z"}],"graph_snapshots":[{"event_id":"sha256:7a26f1933a7dff446823ba313e5031f267a5cb61033b2bef2e5b22fa58777465","target":"graph","created_at":"2026-05-17T23:39:24Z","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":"We construct Fano threefolds with very ample anti-canonical bundle and Picard rank greater than one from cracked polytopes - polytopes whose intersection with a complete fan forms a set of unimodular polytopes - using Laurent inversion; a method developed jointly with Coates-Kasprzyk. We also give constructions of rank one Fano threefolds from cracked polytopes, following work of Christophersen-Ilten and Galkin. We explore the problem of classifying polytopes cracked along a given fan in three dimensions, and classify the unimodular polytopes which can occur as 'pieces' of a cracked polytope.","authors_text":"Thomas Prince","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-04-01T19:31:15Z","title":"From Cracked Polytopes to Fano Threefolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.01077","kind":"arxiv","version":2},"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":"sha256:9974d971c64491e2817f9f4418d9ef0e5b1b5d8c6dbd4881308cedb2cc1aa067","target":"record","created_at":"2026-05-17T23:39:24Z","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":"24a52a386e6a2628025e6d13bcce376b45ec2cc83d6585cdcc79c68ae14f0e66","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-04-01T19:31:15Z","title_canon_sha256":"28cfde70fd8b254fd882b52756e3754728b7523d5a5867a5f21102ff3b019773"},"schema_version":"1.0","source":{"id":"1904.01077","kind":"arxiv","version":2}},"canonical_sha256":"ab73d601d1e3822419fbb7c9269b12f5a6a81f7e0ba0a91e057eaf27dee8f8c7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ab73d601d1e3822419fbb7c9269b12f5a6a81f7e0ba0a91e057eaf27dee8f8c7","first_computed_at":"2026-05-17T23:39:24.495615Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:24.495615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fjpv3ksqWRUby+WMPaWN44e4Sf9qN0cfpQL7uAA6truyWSgZTtGck19AStCeHMdHbiT+CK4txdsI6tDcq4JCDQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:24.497868Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.01077","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9974d971c64491e2817f9f4418d9ef0e5b1b5d8c6dbd4881308cedb2cc1aa067","sha256:7a26f1933a7dff446823ba313e5031f267a5cb61033b2bef2e5b22fa58777465"],"state_sha256":"49c60a54c8943dad48430886139f1c70177bd46182ce845ce02ac32b8d8ce7aa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GIoqA8M3IqzPUzmz+Khsm9+spoh5g4NxiDUUOBAJYwF3MD7EGx9j5l0Y8OoyaiGVpPBVwlVpy0eCif8Zm0vaDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T02:43:33.963332Z","bundle_sha256":"a180d54bbe2cab06235480c78f9833151ba46f2b3146159aa2664fe6f0bb5b4f"}}