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We prove that $W^1_d({C})$ has the expected dimension and that the general element of any irreducible component of $W^1_d({C})$ is primitive if either $g-k+4\\le d\\le g-2$ or $d=g-k+3$ and either $k$ is odd or $C$ is not a double covering of a curve of gonality $k/2$ and genus $k-3$. Even in the latter case we prove the existence of a complete and primitive $g^1_{g-k+3}$."},"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":"1601.02408","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2016-01-11T11:34:04Z","cross_cats_sorted":[],"title_canon_sha256":"b6042f56f1a12478c93672b649caee573daaef6cac4ae571c1a0de769b4beaa1","abstract_canon_sha256":"fa0193494014472dd98e470d766baf14afb2ab0e1adfcb9e9afb2bbe1222d832"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:06.525656Z","signature_b64":"TWmPtyL9O5b8hDGKlJWgpwPf5um7ZF48EnOxandU94F2bETJS5XAPYegAa3jkpnNonowQtyE2kn8xTdHzxR1Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8976afb351237ac459c0593e4a14fd8e8bca3c5a7bdc41e9d9be8c0dcfcdc074","last_reissued_at":"2026-05-18T01:23:06.525089Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:06.525089Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the existence of primitive pencils for smooth curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"E. 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