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As a consequence, we prove a conjecture of Barja and Stoppino on the lower bound of $\\lambda_f$ as an increasing function of $q_f$ in this case, and we also prove a conjecture of Xiao on the ampleness of the direct image of the relative canonical sheaf if $\\lambda_f<4$."},"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":"1311.7271","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-11-28T10:54:55Z","cross_cats_sorted":[],"title_canon_sha256":"68a1351df5c7cd3baa76fd2ce94e3bb58be01d21590039b06a83f7e13808b62a","abstract_canon_sha256":"a47f6da12b3b0ae5de951e31ec1dacf1ba5b84484d50aeac48b16c6f124e1682"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:29.462814Z","signature_b64":"dNUM8GeGf6T4TPjRmuKw/p8pLSUyttxzesqKosY1idKQvRrbzHUDpIjU/aZy6aOpoGl0MPf14MjbbYlSXkwdCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce253f1b0e10862fbb2e107c7c1fb20321553c7e6db0d985fab1012b26dbb225","last_reissued_at":"2026-05-18T02:27:29.462095Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:29.462095Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the slope of hyperelliptic fibrations with positive relative irregularity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Kang Zuo, Xin Lu","submitted_at":"2013-11-28T10:54:55Z","abstract_excerpt":"Let $f:\\, S \\to B$ be a locally non-trivial relatively minimal fibration of hyperelliptic curves of genus $g\\geq 2$ with relative irregularity $q_f$. 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