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We explain, for a given vertex-weighted graph $(G, \\omega)$ in general, how the existence of such $X$ relates the Riemann--Roch formulae for $X$ and $(G, \\omega)$, and also how the existence of such $X$ is related to a conjecture of Caporaso."},"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":"1304.6979","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-04-25T19:12:01Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"bbc70172e2b1bf3990801174d4d9dd3be0143b442d6180e432054c36994f9dff","abstract_canon_sha256":"dd6e0c7ba16848f7ae7c122be246f926c33226c112f07d0fe15757d85244bda0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:02.666440Z","signature_b64":"GFFRT9ANnntxY/sQAZ2myXXggKZr3fWnsZYVtWXTyFuqesb8orxup+XbmovpH+E8a4Rc229qhwi5e5B2jdppCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"10121ef819140f4b09953ab9c40b1d78a97266cb2a7e4f6428013a3efcb67729","last_reissued_at":"2026-05-18T01:37:02.666034Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:02.666034Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rank of divisors on hyperelliptic curves and graphs under specialization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Kazuhiko Yamaki, Shu Kawaguchi","submitted_at":"2013-04-25T19:12:01Z","abstract_excerpt":"Let $(G, \\omega)$ be a hyperelliptic vertex-weighted graph of genus $g \\geq 2$. 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