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We prove that, assuming the Hodge conjecture for the product $S \\times S$, where $S$ is a complex surface, and the finite dimensionality of the Chow motive $h(S)$, there are at most a countable number of decomposable integral polarized Hodge structures, arising from the fibers of a family of smooth projective surfaces. According to the results in [ABB] this is related to a conjecture proving the irrationality of a very general $X$. If $X$ is special, in the sense of B.Hasset, and $F(X) \\simeq S^{[2]}$, with $S$ a K3 surface associated to $X$, then we s"},"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":"1701.05743","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-01-20T10:11:40Z","cross_cats_sorted":[],"title_canon_sha256":"35b05b3dbd585b49a914eb9e8af194d8aae8629ca4e7e8159bed29d53ac5a920","abstract_canon_sha256":"5a65f6eea3f3b884a1581ec827efe38b10dead437f896b09db636cce5c9021ba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:24.874710Z","signature_b64":"Ggt9XGHBeeiczh5AyMeQ0tFtGIfrHE/ERiJPr3pq48sRyWs5gk4v/n5ENxhthKPbwlbyb0Q5/2ZYKp/RRwxsAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"160e71d5ce1d42507a9e4409315ae00bb6cb73b6128d154168d2bba19a8bce59","last_reissued_at":"2026-05-18T00:52:24.874064Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:24.874064Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the rationality and the finite dimensionality of a cubic fourfold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Claudio Pedrini","submitted_at":"2017-01-20T10:11:40Z","abstract_excerpt":"Let $X$ be a cubic fourfold in $P^5_{C}$. We prove that, assuming the Hodge conjecture for the product $S \\times S$, where $S$ is a complex surface, and the finite dimensionality of the Chow motive $h(S)$, there are at most a countable number of decomposable integral polarized Hodge structures, arising from the fibers of a family of smooth projective surfaces. According to the results in [ABB] this is related to a conjecture proving the irrationality of a very general $X$. If $X$ is special, in the sense of B.Hasset, and $F(X) \\simeq S^{[2]}$, with $S$ a K3 surface associated to $X$, then we s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.05743","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1701.05743","created_at":"2026-05-18T00:52:24.874172+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.05743v1","created_at":"2026-05-18T00:52:24.874172+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.05743","created_at":"2026-05-18T00:52:24.874172+00:00"},{"alias_kind":"pith_short_12","alias_value":"CYHHDVOODVBF","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_16","alias_value":"CYHHDVOODVBFA6U6","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_8","alias_value":"CYHHDVOO","created_at":"2026-05-18T12:31:10.602751+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CYHHDVOODVBFA6U6IQETCWXABO","json":"https://pith.science/pith/CYHHDVOODVBFA6U6IQETCWXABO.json","graph_json":"https://pith.science/api/pith-number/CYHHDVOODVBFA6U6IQETCWXABO/graph.json","events_json":"https://pith.science/api/pith-number/CYHHDVOODVBFA6U6IQETCWXABO/events.json","paper":"https://pith.science/paper/CYHHDVOO"},"agent_actions":{"view_html":"https://pith.science/pith/CYHHDVOODVBFA6U6IQETCWXABO","download_json":"https://pith.science/pith/CYHHDVOODVBFA6U6IQETCWXABO.json","view_paper":"https://pith.science/paper/CYHHDVOO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.05743&json=true","fetch_graph":"https://pith.science/api/pith-number/CYHHDVOODVBFA6U6IQETCWXABO/graph.json","fetch_events":"https://pith.science/api/pith-number/CYHHDVOODVBFA6U6IQETCWXABO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CYHHDVOODVBFA6U6IQETCWXABO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CYHHDVOODVBFA6U6IQETCWXABO/action/storage_attestation","attest_author":"https://pith.science/pith/CYHHDVOODVBFA6U6IQETCWXABO/action/author_attestation","sign_citation":"https://pith.science/pith/CYHHDVOODVBFA6U6IQETCWXABO/action/citation_signature","submit_replication":"https://pith.science/pith/CYHHDVOODVBFA6U6IQETCWXABO/action/replication_record"}},"created_at":"2026-05-18T00:52:24.874172+00:00","updated_at":"2026-05-18T00:52:24.874172+00:00"}