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arXiv · 2009.13421

On the proportion of transverse-free plane curves

Shamil Asgarli·Brian Freidin
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Abstract

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.

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Shamil Asgarli, Brian Freidin. 2020-09-28. On the proportion of transverse-free plane curves. https://arxiv.org/abs/2009.13421

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