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For each $m$ in the semigroup \\[\n  \\mathbf{N}(V_\\bullet)=\\{m\\in\\mathbb{N}\\mid V_m\\ne 0\\},\\] the linear series $V_m$ defines a rational map \\[ \\phi_m\\colon X\\dashrightarrow Y_m\\subseteq\\mathbb{P}(V_m), \\] where $Y_m$ d"},"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":"1903.05967","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-03-14T13:08:35Z","cross_cats_sorted":[],"title_canon_sha256":"398c616af86826880be4da1ba88e01fde6059d1258c6b07fbf3c31b8e303fb62","abstract_canon_sha256":"7927d53a8af18ae6e887f4afc45c2789fbc236080b6bd5c42e3dd1284cfd2aa3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:15.702310Z","signature_b64":"w+n4JpPcJZUoqSRYjgp74tGXSC9W0kmIcwEczthyia2qa66lNdybwzWLN19VcZs90cdc7KyBURNlpXad9yYXCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"97d678734d62af21081ebe933a5dc53b8f435872943fda004f8f792f60afa1cc","last_reissued_at":"2026-05-17T23:51:15.701923Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:15.701923Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic constructions and invariants of graded linear series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Chih-Wei Chang, Shin-Yao Jow","submitted_at":"2019-03-14T13:08:35Z","abstract_excerpt":"Let $X$ be a complete variety of dimension $n$ over an algebraically closed field $\\mathbf{K}$. 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