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In this paper, we describe all Borel subgroups of the complex Cremona group ${{\\rm Bir}({\\mathbb P}^2)}$ up to conjugation, proving in particular that they are not necessarily conjugate. More precisely, we prove that ${{\\rm Bir}({\\mathbb P}^2)}$ admits Borel subgroups of any rank $r \\in \\{ 0,1,2 \\}$ and that all Borel subgroups of rank $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":"2107.14353","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2021-07-29T22:22:16Z","cross_cats_sorted":[],"title_canon_sha256":"9ddcafaaefe1cbc8727edb27201769e60d1837d2e86c9cd1048b345256793604","abstract_canon_sha256":"e3cca1730a02da11f70184dfe467656ef19d2dcb9b3661ec2419bb8d08a4b7a9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:59:57.350517Z","signature_b64":"PKfJtxZOOgfdTXVWjsNm3we0iCz4QyaseES166+FQKNmNNOxbAG2/172mQkAd8X6bH8I4981/djnWnHJbV0VDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c0d00361395afefbc44a2a2ee67c15b50052ffca327b98d84a48a5923fcdc70","last_reissued_at":"2026-07-05T04:59:57.350136Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:59:57.350136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Borel subgroups of the plane Cremona group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Isac Hed\\'en, Jean-Philippe Furter","submitted_at":"2021-07-29T22:22:16Z","abstract_excerpt":"It is well known that all Borel subgroups of a linear algebraic group are conjugate. This result also holds for the automorphism group ${{\\mathrm{Aut}}} (\\mathbb A^2)$ of the affine plane \\cite{BerestEshmatovEshmatov2016} (see also \\cite{FurterPoloni2018}). In this paper, we describe all Borel subgroups of the complex Cremona group ${{\\rm Bir}({\\mathbb P}^2)}$ up to conjugation, proving in particular that they are not necessarily conjugate. More precisely, we prove that ${{\\rm Bir}({\\mathbb P}^2)}$ admits Borel subgroups of any rank $r \\in \\{ 0,1,2 \\}$ and that all Borel subgroups of rank $r \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.14353","kind":"arxiv","version":3},"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/2107.14353/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":"2107.14353","created_at":"2026-07-05T04:59:57.350192+00:00"},{"alias_kind":"arxiv_version","alias_value":"2107.14353v3","created_at":"2026-07-05T04:59:57.350192+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2107.14353","created_at":"2026-07-05T04:59:57.350192+00:00"},{"alias_kind":"pith_short_12","alias_value":"BQGQANQTSWX6","created_at":"2026-07-05T04:59:57.350192+00:00"},{"alias_kind":"pith_short_16","alias_value":"BQGQANQTSWX67PCE","created_at":"2026-07-05T04:59:57.350192+00:00"},{"alias_kind":"pith_short_8","alias_value":"BQGQANQT","created_at":"2026-07-05T04:59:57.350192+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BQGQANQTSWX67PCEUKRO4Z6BLN","json":"https://pith.science/pith/BQGQANQTSWX67PCEUKRO4Z6BLN.json","graph_json":"https://pith.science/api/pith-number/BQGQANQTSWX67PCEUKRO4Z6BLN/graph.json","events_json":"https://pith.science/api/pith-number/BQGQANQTSWX67PCEUKRO4Z6BLN/events.json","paper":"https://pith.science/paper/BQGQANQT"},"agent_actions":{"view_html":"https://pith.science/pith/BQGQANQTSWX67PCEUKRO4Z6BLN","download_json":"https://pith.science/pith/BQGQANQTSWX67PCEUKRO4Z6BLN.json","view_paper":"https://pith.science/paper/BQGQANQT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2107.14353&json=true","fetch_graph":"https://pith.science/api/pith-number/BQGQANQTSWX67PCEUKRO4Z6BLN/graph.json","fetch_events":"https://pith.science/api/pith-number/BQGQANQTSWX67PCEUKRO4Z6BLN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BQGQANQTSWX67PCEUKRO4Z6BLN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BQGQANQTSWX67PCEUKRO4Z6BLN/action/storage_attestation","attest_author":"https://pith.science/pith/BQGQANQTSWX67PCEUKRO4Z6BLN/action/author_attestation","sign_citation":"https://pith.science/pith/BQGQANQTSWX67PCEUKRO4Z6BLN/action/citation_signature","submit_replication":"https://pith.science/pith/BQGQANQTSWX67PCEUKRO4Z6BLN/action/replication_record"}},"created_at":"2026-07-05T04:59:57.350192+00:00","updated_at":"2026-07-05T04:59:57.350192+00:00"}