{"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 $e_0, e_1, e_2 \\in \\mathbb{F}_q$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.04442","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"}