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I show that, for every integer $m$ such that $m_u \\leq m \\leq M_u$, there exists a scheme with Hilbert function $u$ and Castelnuovo-Mumford regularity $m$. As a consequence, the analogous algebraic result for an O-sequence $f$ and homogeneous polynomial ideals over $K$ with Hilbert function $f$ holds too.\n  Although this result does not need an"},"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":"1901.10974","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-01-30T17:57:31Z","cross_cats_sorted":[],"title_canon_sha256":"83d90439086240c825096323b7147846ee3e9ec125b481914b915b53ea793e8f","abstract_canon_sha256":"dd42fb301a4d5361145e260861ca005c0d758abe852a80be679a02770662d312"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:05.621737Z","signature_b64":"jo5QnR+DFLodmUga7Ys/bFxGehXgBDBpi+pY546nvwg2X3aUqeT0qyWN/cKGZ0k9i7JMDw6QSv0H1zzxm8kRDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"69c851496aae1770585829b7babaccf81352026ba97ed84a12637dd388927fb6","last_reissued_at":"2026-05-17T23:55:05.621083Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:05.621083Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The range of all regularities for polynomial ideals with a given Hilbert function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Francesca Cioffi","submitted_at":"2019-01-30T17:57:31Z","abstract_excerpt":"Given the Hilbert function $u$ of a closed subscheme of a projective space over an infinite field $K$, let $m_u$ and $M_u$ be, respectively, the minimum and the maximum among all the Castelnuovo-Mumford regularities of schemes with Hilbert function $u$. I show that, for every integer $m$ such that $m_u \\leq m \\leq M_u$, there exists a scheme with Hilbert function $u$ and Castelnuovo-Mumford regularity $m$. As a consequence, the analogous algebraic result for an O-sequence $f$ and homogeneous polynomial ideals over $K$ with Hilbert function $f$ holds too.\n  Although this result does not need an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.10974","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1901.10974","created_at":"2026-05-17T23:55:05.621176+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.10974v1","created_at":"2026-05-17T23:55:05.621176+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.10974","created_at":"2026-05-17T23:55:05.621176+00:00"},{"alias_kind":"pith_short_12","alias_value":"NHEFCSLKVYLX","created_at":"2026-05-18T12:33:24.271573+00:00"},{"alias_kind":"pith_short_16","alias_value":"NHEFCSLKVYLXAWCY","created_at":"2026-05-18T12:33:24.271573+00:00"},{"alias_kind":"pith_short_8","alias_value":"NHEFCSLK","created_at":"2026-05-18T12:33:24.271573+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/NHEFCSLKVYLXAWCYFG33VOWM7A","json":"https://pith.science/pith/NHEFCSLKVYLXAWCYFG33VOWM7A.json","graph_json":"https://pith.science/api/pith-number/NHEFCSLKVYLXAWCYFG33VOWM7A/graph.json","events_json":"https://pith.science/api/pith-number/NHEFCSLKVYLXAWCYFG33VOWM7A/events.json","paper":"https://pith.science/paper/NHEFCSLK"},"agent_actions":{"view_html":"https://pith.science/pith/NHEFCSLKVYLXAWCYFG33VOWM7A","download_json":"https://pith.science/pith/NHEFCSLKVYLXAWCYFG33VOWM7A.json","view_paper":"https://pith.science/paper/NHEFCSLK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.10974&json=true","fetch_graph":"https://pith.science/api/pith-number/NHEFCSLKVYLXAWCYFG33VOWM7A/graph.json","fetch_events":"https://pith.science/api/pith-number/NHEFCSLKVYLXAWCYFG33VOWM7A/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NHEFCSLKVYLXAWCYFG33VOWM7A/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NHEFCSLKVYLXAWCYFG33VOWM7A/action/storage_attestation","attest_author":"https://pith.science/pith/NHEFCSLKVYLXAWCYFG33VOWM7A/action/author_attestation","sign_citation":"https://pith.science/pith/NHEFCSLKVYLXAWCYFG33VOWM7A/action/citation_signature","submit_replication":"https://pith.science/pith/NHEFCSLKVYLXAWCYFG33VOWM7A/action/replication_record"}},"created_at":"2026-05-17T23:55:05.621176+00:00","updated_at":"2026-05-17T23:55:05.621176+00:00"}