论文标题
在无横向平面曲线的比例
On the proportion of transverse-free plane curves
论文作者
论文摘要
我们研究了在有限字段$ \ mathbb {f} _q $上的平滑平面曲线的渐近比例，这些曲线与$ \ mathbb {f} _q $相切。这部分回答了查尔斯·法夫尔（Charles Favre）提出的一个问题。我们的技术包括Poonen的Bertini定理和Schrijver定理的应用，以在常规的两部分图中进行完美匹配。我们的主要定理意味着，$ \ mathbb {f} _q $上的随机平滑平面曲线允许横向$ \ mathbb {f} _q $ - line，概率很高。
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.
