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Sharp Bertini Theorem for Plane Curves over Finite Fields

Part of: Arithmetic algebraic geometry Curves Projective and enumerative geometry

Published online by Cambridge University Press:  07 January 2019

Shamil Asgarli
Shamil Asgarli*
Affiliation:
Department of Mathematics, Brown University, Providence, RI 02912, USA Email: shamil_asgarli@brown.edu
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Abstract

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We prove that if $C$ is a reflexive smooth plane curve of degree $d$ defined over a finite field $\mathbb{F}_{q}$ with $d\leqslant q+1$, then there is an $\mathbb{F}_{q}$-line $L$ that intersects $C$ transversely. We also prove the same result for non-reflexive curves of degree $p+1$ and $2p+1$ when $q=p^{r}$.

Keywords

Bertini theoremtransversalityfinite field

MSC classification

Primary: 14H50: Plane and space curves
Secondary: 11G20: Curves over finite and local fields 14N05: Projective techniques
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 2 , June 2019 , pp. 223 - 230
Copyright
© Canadian Mathematical Society 2018 

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