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.

中文翻译：

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关于无横向平面曲线的比例

我们研究有限域上的光滑平面曲线的渐近比例 ${\mathbb{F}}_q$ 与定义在上面的每条线相切 ${\mathbb{F}}_q$。这部分地回答了查尔斯·法弗（Charles Favre）提出的问题。我们的技术包括Poonen的Bertini定理和Schrijver定理在正则二部图中的完美匹配上的应用。我们的主要定理表明，在${\mathbb{F}}_q$ 承认横向 ${\mathbb{F}}_q$线的可能性很高。