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The corresponding result for length 3 is well-known and states that there is essentially one solution, namely the one corresponding to the standard exceptional collection on the surface $\\mathbb{P}^2$. This was essentially proven by Markov in 1879. It turns out that in the length 4 case, there is one special solution which corresponds to $\\mathbb{P}^1\\times\\mathbb{P}^1$ whereas the other solutions are obtained from $\\mathbb{P}^2$ "},"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":"1607.04246","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2016-07-14T18:53:56Z","cross_cats_sorted":[],"title_canon_sha256":"cdad06d19540addbd0604a3acf60f1bdc22862fa567826066a5e8acda13b31a8","abstract_canon_sha256":"555f061b664a9e875461992c0a221d23e110c17f9ccf29d4cb1332d1d03799a1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:03.772282Z","signature_b64":"xm1Yl+kmFjpPpVhS3zUr8RsAZFJ7jBhtoq4gCOjghGR6mB2SUURMYB/3fC30kWvZ9CXQrMrQt6KKMBdsru2sAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"893f8071b423c156cb3347d5db973112307400d13f5a8190d00db5f02bf5cef3","last_reissued_at":"2026-05-18T01:11:03.771813Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:03.771813Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On an analogue of the Markov equation for exceptional collections of length 4","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Louis de Thanhoffer de Volcsey, Michel Van den Bergh","submitted_at":"2016-07-14T18:53:56Z","abstract_excerpt":"We classify the solutions to a system of equations, introduced by Bondal, which encode numerical constraints on full exceptional collections of length 4 on surfaces. 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