{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:TKKCSGULFM2NGPC5T5UCBR7XPO","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":"3ca24167ed5022e37ac52f03de8f58149cc17242ea23fd45683dc5e9dcd4ce75","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-03-25T15:13:21Z","title_canon_sha256":"f179cc4c431ffa4bea003e2a084c9155446b789eef0b3726c358ace6762c3c16"},"schema_version":"1.0","source":{"id":"1203.5505","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.5505","created_at":"2026-05-18T03:59:14Z"},{"alias_kind":"arxiv_version","alias_value":"1203.5505v1","created_at":"2026-05-18T03:59:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.5505","created_at":"2026-05-18T03:59:14Z"},{"alias_kind":"pith_short_12","alias_value":"TKKCSGULFM2N","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"TKKCSGULFM2NGPC5","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"TKKCSGUL","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:de571b32cacf2025fa628314d4649778969a0d5ed0020df29dbdce12a2187bc4","target":"graph","created_at":"2026-05-18T03:59:14Z","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 investigate the triangle singularity $f=x^a+y^b+z^c$, or $S=k[x,y,z]/(f)$, attached to a weighted projective line $X$ given by the weight triple $(a,b,c)$. We investigate the stable category of vector bundles on $X$ obtained from the vector bundles by factoring out all line bundles. This category is triangulated and has Serre duality. It is, moreover, naturally equivalent to the stable category of graded maximal Cohen-Macaulay modules over $S$ (or matrix factorizations of $f$), and then by results of Buchweitz and Orlov to the graded singularity category of $f$. We show that this category i","authors_text":"Dirk Kussin, Hagen Meltzer, Helmut Lenzing","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-03-25T15:13:21Z","title":"Triangle singularities, ADE-chains, and weighted projective lines"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.5505","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:fa337aed03e22097550cbe18e4fcc29733db0ff5172af42ea94b11949d80e23a","target":"record","created_at":"2026-05-18T03:59:14Z","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":"3ca24167ed5022e37ac52f03de8f58149cc17242ea23fd45683dc5e9dcd4ce75","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-03-25T15:13:21Z","title_canon_sha256":"f179cc4c431ffa4bea003e2a084c9155446b789eef0b3726c358ace6762c3c16"},"schema_version":"1.0","source":{"id":"1203.5505","kind":"arxiv","version":1}},"canonical_sha256":"9a94291a8b2b34d33c5d9f6820c7f77b89eab29388f1e078d2c0ef3139f1a5bd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9a94291a8b2b34d33c5d9f6820c7f77b89eab29388f1e078d2c0ef3139f1a5bd","first_computed_at":"2026-05-18T03:59:14.902240Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:59:14.902240Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6Q+3UZQoZwwP4N60wRIBF1cweXwsXyLOiabfcXEP+nALz+lE8NchhhYelHcmyNn5s9rtXVcX+Rj2P9ZOAfuMAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:59:14.903073Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.5505","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fa337aed03e22097550cbe18e4fcc29733db0ff5172af42ea94b11949d80e23a","sha256:de571b32cacf2025fa628314d4649778969a0d5ed0020df29dbdce12a2187bc4"],"state_sha256":"58429891c5ec58104d92795568fb975804d98d7429416a8bd72e575967a611fb"}