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We give sufficient conditions for having such inequality, depending on the genus and gonality of $C$ and the number of certain exceptional curves on $S$. We illustrate how these results can be used for constructing fibrations with the desired property. For fibrations of genus $11\\le g\\le 49$ we prove the inequality: $$ 6(g-1) +4 -4\\sqrt g \\le K_f^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":"1001.0177","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2009-12-31T20:00:37Z","cross_cats_sorted":[],"title_canon_sha256":"cd25d1f2de59b7a6de3c808712cd295669abc67ea54ea5280d1383a8a64595e2","abstract_canon_sha256":"6a1ad439b997f566dd879373240f9f48d36f2abe69d26ed85b36d8deb88b6f04"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:12.397090Z","signature_b64":"z2vzBemKuIOE52Pgv6AYdRdk6RjcLBhw5zi4nJJvasOGuGULdClLRUtlnrGWw/HH8XIzxDO2xrIyY0GOwBdnAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e9d1d4ee9785846297bb6bebc9d04b58e899b2172ca8e616d06751ed68e0b70c","last_reissued_at":"2026-05-18T04:42:12.396435Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:12.396435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the slope of relatively minimal fibrations on rational complex surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Abel Castorena, Alexis G. 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