Transverse lines to surfaces over finite fields

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 $q\ge c\mathrm{deg}(S)$, 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 $q\ge \mathrm{deg}(S{)}^2$.

24 pages, final version

References in corpus (2)

• Sharp Bertini theorem for plane curves over finite fields
• Numbers of points of hypersurfaces without lines over finite fields

Cited by in corpus (1)

• Transverse linear subspaces to hypersurfaces over finite fields