Smooth Surfaces with Maximal Lines

2024/06/22 by Janet Page, Page, Janet, T. P. Ryan +3
Engineering · Mathematics · #14G17 #14J25 #14N15 #51E12 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #primary: 14N10 #secondary: 13A35

paper · pdf · doi:10.48550/arxiv.2406.15868

openalex publication_date 2024/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that a smooth projective surface of degree d in \mathbb P³ contains at most d²(d²-3d+3) lines. We characterize the surfaces containing exactly d²(d²-3d+3) lines: these occur only in prime characterize p and, up to choice of projective coordinates, are cut out by equations of the form x^p^e+1+y^p^e+1+z^p^e+1+ w^p^e+1 = 0.

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