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If the Albanese morphism $a_X:X\\to \\mathrm{Alb}_X$ is generically finite onto its image, we formulate a constant $c(X,L)\\ge 0$ for a very ample line bundle $L$ on $\\mathrm{Alb}_X$ such that $c(X,L)=0$ if and only if $\\dim \\mathrm{Alb}_X=2$ and $a_X: X\\to \\mathrm{Alb}_X$ is a double cover. A refined Severi inequality $$K^2_X\\ge (4+{\\rm min}\\{\\,c(X,L),\\,\\frac{1}{3}\\,\\})\\chi(\\mathcal{O}_X)$$ is proved. Then we prove that $K^2_X=4\\chi(\\mathcal{O}_X)$ if and only if t"},"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":"1908.01933","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-08-06T02:31:03Z","cross_cats_sorted":[],"title_canon_sha256":"e8c117b1742b7abbd3c244dd29693ed243adb7b0fde66e19f33b7152d3c0ddc7","abstract_canon_sha256":"4e79ab7487c5017490312aec9c0d54d8fd559ed6daa2c08d7c6e406a0c476537"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:05:24.826677Z","signature_b64":"GeT+HeR+JqTPGJegrgLxs3EIF8ccnVoT3aGuXIC3BdqoQF1NPh3TnxSD1eO+qO4brykzV60omLfXD3u/gFgYBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"700c508a03ac95a510b62e25c79cb5b59bbc1b7bf877108e8b36063ef66b7c05","last_reissued_at":"2026-07-05T00:05:24.826289Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:05:24.826289Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Surfaces on the Severi line in positive characteristics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Mingshuo Zhou, Xiaotao Sun, Yi Gu","submitted_at":"2019-08-06T02:31:03Z","abstract_excerpt":"Let $X$ be a minimal surface of general type over an algebraically closed field $\\mathbf{k}$ of $\\mathrm{char}.(\\mathbf{k})=p\\ge 0$. If the Albanese morphism $a_X:X\\to \\mathrm{Alb}_X$ is generically finite onto its image, we formulate a constant $c(X,L)\\ge 0$ for a very ample line bundle $L$ on $\\mathrm{Alb}_X$ such that $c(X,L)=0$ if and only if $\\dim \\mathrm{Alb}_X=2$ and $a_X: X\\to \\mathrm{Alb}_X$ is a double cover. A refined Severi inequality $$K^2_X\\ge (4+{\\rm min}\\{\\,c(X,L),\\,\\frac{1}{3}\\,\\})\\chi(\\mathcal{O}_X)$$ is proved. 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