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As an application we have \\[ {\\rm{vol}}(X)\\ge\\frac{4}{3}p_g(X)-\\frac{10}{3} \\] for all projective $3$-folds $X$ of general type which answers Question 1.4 of [J. A. Chen, M. Chen, C. Jiang, \"The Noether inequality for algebraic threefolds\", arXiv:1803.05553] about Noether inequality for $X$ with $5\\le p_g(X)\\le 26$."},"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":"1805.01323","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2018-05-03T14:29:35Z","cross_cats_sorted":[],"title_canon_sha256":"8a1f864a37f310f4ac2101a5d38f745a64dd06541f8478b4571774e55f7ac662","abstract_canon_sha256":"780552cae4d2694c54e9d3feed94f698fe85dc758f41b8aa170c6bd08a60265f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:49.791513Z","signature_b64":"W+n+9TfqtqWbvky2vlIr5Cgo9jFvD2HuaX1rzu/LqGupGDQT8NQcLwaTl9LYDttPljOHOj3vMF88Tk5QWfvECg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d4287c8e99e807f04838efb1ddce8ae56649bf5e25bf77fc9dbf52f9f7fcb9ab","last_reissued_at":"2026-05-18T00:16:49.790770Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:49.790770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global log canonical thresholds of minimal $(1,2)$-surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"In-Kyun Kim, Joonyeong Won, YongJoo Shin","submitted_at":"2018-05-03T14:29:35Z","abstract_excerpt":"Let $S$ be a minimal surface of general type with $p_g(S)=2$ and $K^2_S=1$, so called by a minimal $(1,2)$-surface. Then we obtain that the global log canonical threshold of the surface $S$ via $K_S$ is greater than equal to $\\frac{1}{2}$. As an application we have \\[ {\\rm{vol}}(X)\\ge\\frac{4}{3}p_g(X)-\\frac{10}{3} \\] for all projective $3$-folds $X$ of general type which answers Question 1.4 of [J. A. Chen, M. Chen, C. 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