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In particular, we will prove that any nonlinear component $\\mathcal{G} \\subseteq \\mathcal{P} \\cap \\mathcal{S} $ is a smooth classical curve of degree $n\\leqslant d$ attaining the St\\\"ohr-Voloch bound $$ \\# \\mathcal{G}(\\mathbb{F}_q) \\leqslant \\frac{1}{2} n(n+q-1) - \\frac{1}{2} i(n-2), $$ with $i"},"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":"1804.04442","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2018-04-12T11:46:50Z","cross_cats_sorted":[],"title_canon_sha256":"0a1b13ada4005f61b6af8a5fd7778bbe7ad8ce917538612ae73baf8d2013af3b","abstract_canon_sha256":"338f4dc7962d47ee6a0b87d75d9e2c8bf7edaf632e65e5006c6c92238c842272"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:18:37.539268Z","signature_b64":"iI+OGf+Jin7TNgj+ryS18IH648dAUkI/avMi2ymhRLqS3X54zlfathAROl2k7Ufes6UKQgkE7B3NigGtq7wNBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c0ae21439adeddc3374bf0959b36f3a464a47f21a38b7d1cf4a62af8f96570d2","last_reissued_at":"2026-05-18T00:18:37.538652Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:18:37.538652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Plane sections of Fermat surfaces over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"G. Cook, H. Borges, M. Coutinho","submitted_at":"2018-04-12T11:46:50Z","abstract_excerpt":"In this paper, we characterize all curves over $\\mathbb{F}_q$ arising from a plane section $$ \\mathcal{P} : X_3-e_0X_0-e_1X_1-e_2X_2 = 0 $$ of the Fermat surface $$ \\mathcal{S} : X_0^d + X_1^d + X_2^d +X_3^d = 0, $$ where $q = p^{h} = 2d+1$ is a prime power, $p >3$, and $e_0, e_1, e_2 \\in \\mathbb{F}_q$. 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