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arXiv · 2406.15868

Smooth Surfaces with Maximal Lines

Janet Page·Tim Ryan·Karen E. Smith
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Abstract

We prove that a smooth projective surface of degree $d$ in $\mathbb P^3$ contains at most $d^2(d^2-3d+3)$ lines. We characterize the surfaces containing exactly $d^2(d^2-3d+3)$ 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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Janet Page, Tim Ryan, Karen E. Smith. 2024-06-22. Smooth Surfaces with Maximal Lines. https://arxiv.org/abs/2406.15868

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