A plane curve of degree is called if every -line in the plane meets at some -point. We prove that the proportion of blocking curves among those of degree is when and . We also show that the same conclusion holds for smooth curves under the somewhat weaker condition and . Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent. Extending a classical result on the number of -roots of random polynomials, we find that the limiting distribution of the number of -points in the intersection of a random plane curve and a fixed -line is Poisson with mean 1. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of -points contained in a union of lines for .

中文翻译：

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有限域上的大多数平面曲线都不会阻塞

如果平面上的每一条线都交于某个点，则称为度平面曲线。我们证明当 和 时，阻塞曲线在度数曲线中所占的比例为 。我们还表明，在稍弱的条件 和 下，相同的结论也适用于平滑曲线。此外，随机平面曲线光滑且分别阻塞的两个事件被证明是渐近独立的。将经典结果推广到随机多项式的根数上，我们发现随机平面曲线与固定线相交处的点数的极限分布是均值为 1 的泊松分布。我们还提出了一个显式的分块曲线比例的公式，涉及 的线并集中包含的 - 点数量的统计。