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arXiv · 1903.08845

Transverse lines to surfaces over finite fields

Shamil Asgarli·Lian Duan·Kuan-Wen Lai
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Abstract

We prove that if $S$ is a smooth reflexive surface in $\mathbb{P}^3$ defined over a finite field $\mathbb{F}_q$, then there exists an $\mathbb{F}_q$-line meeting $S$ transversely provided that $q\geq c\operatorname{deg}(S)$, where $c=\frac{3+\sqrt{17}}{4}\approx 1.7808$. Without the reflexivity hypothesis, we prove the existence of a transverse $\mathbb{F}_q$-line for $q\geq \operatorname{deg}(S)^2$.

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Shamil Asgarli, Lian Duan, Kuan-Wen Lai. 2019-03-21. Transverse lines to surfaces over finite fields. https://doi.org/10.1007/s00229-020-01200-7

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