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Eisenbud has shown that a decomposition D = A + B such that A and B have at least two sections gives rise to determinantal equations (and corresponding syzygies) in I_X; and conjectured that if I_2 is generated by quadrics of rank at most 4, then the last nonvanishing b_(i,i+1) is a consequence of such equations. We describe obstruction"},"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":"1012.0933","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-12-04T17:03:03Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"8a1743312b73b39708f2ee1fa7a391e674c48a75162289b9a410f05e4aab22e1","abstract_canon_sha256":"7a23768e506b5968c6c16d900e6bcdbc3df86b6481decc4176e771d3668522ad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:47:53.301703Z","signature_b64":"XjJtwl5l8xhjlEvQHqf73VWQrUUKYv2ufSWS94QnakrCrJ7oq3CuOPxzrFrFg5yy6GktWDKF3jfwu2bmx6AgDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b87cab5b20db43e7ba3103a4c4bb91bc2c443b2a048551155431855e5c16daf7","last_reissued_at":"2026-05-18T02:47:53.301146Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:47:53.301146Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"High rank linear syzygies on low rank quadrics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.AG","authors_text":"Hal Schenck, Mike Stillman","submitted_at":"2010-12-04T17:03:03Z","abstract_excerpt":"We study the linear syzygies of a homogeneous ideal I in a polynomial ring S, focusing on the graded betti numbers b_(i,i+1) = dim_k Tor_i(S/I, k)_(i+1). For a variety X and divisor D with S = Sym(H^0(D)*), what conditions on D ensure that b_(i,i+1) is nonzero? Eisenbud has shown that a decomposition D = A + B such that A and B have at least two sections gives rise to determinantal equations (and corresponding syzygies) in I_X; and conjectured that if I_2 is generated by quadrics of rank at most 4, then the last nonvanishing b_(i,i+1) is a consequence of such equations. 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