On the proportion of transverse-free plane curves

We study the asymptotic proportion of smooth plane curves over a finite field \(\mathbb{F}_q\) which are tangent to every line defined over \(\mathbb{F}_q\). This partially answers a question raised by Charles Favre. Our techniques include applications of Poonen's Bertini theorem and Schrijver&...

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Bibliographic Details
Published in:arXiv.org 2020-09
Main Authors: Asgarli, Shamil, Freidin, Brian
Format: Article
Language:English
Subjects:
Curves
Fields (mathematics)
Theorems
Online Access:Get full text
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Summary:We study the asymptotic proportion of smooth plane curves over a finite field \(\mathbb{F}_q\) which are tangent to every line defined over \(\mathbb{F}_q\). This partially answers a question raised by Charles Favre. Our techniques include applications of Poonen's Bertini theorem and Schrijver's theorem on perfect matchings in regular bipartite graphs. Our main theorem implies that a random smooth plane curve over \(\mathbb{F}_q\) admits a transverse \(\mathbb{F}_q\)-line with very high probability.
ISSN:2331-8422