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Then the number of incidences between $P$ and $L$ is $$ I(P,L)=O\\left(m^{1/2}n^{1/2}D^{1/2} + m^{2/3}\\min\\{n,D^{2}\\}^{1/3}s^{1/3} + m + n\\right). $$ When $d=3$, this improves the bound of Guth and Katz~\\cite{GK2} for this special case, when $D$ is not too large.\n  "},"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":"1502.01670","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-01-13T05:30:37Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"120fd3dd388ce2c8b441b836e2a82ffa5c2ceff016294e29a3519d6b7c8a6d46","abstract_canon_sha256":"99881743db3bc2c84fe09ca144c9f2f269c1acb88058f051f71bf17769f4918c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:59:37.588559Z","signature_b64":"KsSSo0fSJxN9i3/1dWAHOEJ9bCIc77tHntPxCUMv5X7xQnZG1cHy6zINrvrwMrEj36O4gN2wbB15SiPGAvKgCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"23386d9a7ad35fe7fc69fc8ef3a0d783aa642263a4eeaff2439a3dd0a95b027f","last_reissued_at":"2026-05-18T01:59:37.587791Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:59:37.587791Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Incidences between points and lines on a two-dimensional variety","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AG","authors_text":"Micha Sharir, Noam Solomon","submitted_at":"2015-01-13T05:30:37Z","abstract_excerpt":"We present a direct and fairly simple proof of the following incidence bound: Let $P$ be a set of $m$ points and $L$ a set of $n$ lines in ${\\mathbb R}^d$, for $d\\ge 3$, which lie in a common algebraic two-dimensional surface of degree $D$ that does not contain any 2-flat, so that no 2-flat contains more than $s \\le D$ lines of $L$. Then the number of incidences between $P$ and $L$ is $$ I(P,L)=O\\left(m^{1/2}n^{1/2}D^{1/2} + m^{2/3}\\min\\{n,D^{2}\\}^{1/3}s^{1/3} + m + n\\right). $$ When $d=3$, this improves the bound of Guth and Katz~\\cite{GK2} for this special case, when $D$ is not too large.\n  "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01670","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":"1502.01670","created_at":"2026-05-18T01:59:37.587909+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.01670v1","created_at":"2026-05-18T01:59:37.587909+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.01670","created_at":"2026-05-18T01:59:37.587909+00:00"},{"alias_kind":"pith_short_12","alias_value":"EM4G3GT22NP6","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_16","alias_value":"EM4G3GT22NP6P7DJ","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_8","alias_value":"EM4G3GT2","created_at":"2026-05-18T12:29:19.899920+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/EM4G3GT22NP6P7DJ7SHPHIGXQO","json":"https://pith.science/pith/EM4G3GT22NP6P7DJ7SHPHIGXQO.json","graph_json":"https://pith.science/api/pith-number/EM4G3GT22NP6P7DJ7SHPHIGXQO/graph.json","events_json":"https://pith.science/api/pith-number/EM4G3GT22NP6P7DJ7SHPHIGXQO/events.json","paper":"https://pith.science/paper/EM4G3GT2"},"agent_actions":{"view_html":"https://pith.science/pith/EM4G3GT22NP6P7DJ7SHPHIGXQO","download_json":"https://pith.science/pith/EM4G3GT22NP6P7DJ7SHPHIGXQO.json","view_paper":"https://pith.science/paper/EM4G3GT2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.01670&json=true","fetch_graph":"https://pith.science/api/pith-number/EM4G3GT22NP6P7DJ7SHPHIGXQO/graph.json","fetch_events":"https://pith.science/api/pith-number/EM4G3GT22NP6P7DJ7SHPHIGXQO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EM4G3GT22NP6P7DJ7SHPHIGXQO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EM4G3GT22NP6P7DJ7SHPHIGXQO/action/storage_attestation","attest_author":"https://pith.science/pith/EM4G3GT22NP6P7DJ7SHPHIGXQO/action/author_attestation","sign_citation":"https://pith.science/pith/EM4G3GT22NP6P7DJ7SHPHIGXQO/action/citation_signature","submit_replication":"https://pith.science/pith/EM4G3GT22NP6P7DJ7SHPHIGXQO/action/replication_record"}},"created_at":"2026-05-18T01:59:37.587909+00:00","updated_at":"2026-05-18T01:59:37.587909+00:00"}