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Moreover, equality holds only if the general fibre $F$ of the Albanese morphism of $X$ is a smooth minimal surface of general type satisfying $p_g(F)=3,q(F)=0$ and $K_F^2=36$, and the canonical map of $F$ has degree $36$. This result improves the lower bound on $p_g(X)$ previously obtained by Jin-Xing Cai~\\cite{Cai08}.\n  As a consequence, we show that if the canonical degree is bigger than $64$, then the genera"},"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":"2606.31170","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-06-30T06:00:07Z","cross_cats_sorted":[],"title_canon_sha256":"89df346b89ce1411d3ef51327c334caf69e5e07998d663c0015df9f8d609fa88","abstract_canon_sha256":"e0c23a81be27ec6ea419e7a82092ccabb361f7f5a846c7c06a8739b8168c7c13"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-01T01:17:31.080910Z","signature_b64":"g9fbRh0+vy/GNtRWFD0LIr7HplxK3Tdl/R/ciP7GjzxzkJqpfD8yQ/t9kyMLMUv332aNsP2l08CVybJM4+8sAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99734dd0a6365bc84da084ad421f42430714a9c8132f2bce370f695f5225623a","last_reissued_at":"2026-07-01T01:17:31.080526Z","signature_status":"signed_v1","first_computed_at":"2026-07-01T01:17:31.080526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the canonical degree of a Gorenstein minimal threefold of general type","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Jiabin Du, Yong Hu","submitted_at":"2026-06-30T06:00:07Z","abstract_excerpt":"Let $X$ be a Gorenstein minimal $3$-fold of general type whose canonical map is generically finite. 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