We study incidence problems involving points and curves in $R^3$. The current (and in fact only viable) approach to such problems, pioneered by Guth and Katz, requires a variety of tools from algebraic geometry, most notably (i) the polynomial partitioning technique, and (ii) the study of algebraic surfaces that are ruled by lines or, in more recent studies, by algebraic curves of some constant degree. By exploiting and refining these tools, we obtain new and improved bounds for point-curve incidence problems in $R^3$. Incidences of this kind have been considered in several previous studies, starting with Guth and Katz's work on points and lines. Our results, which are based on the work of Guth and Zahl concerning surfaces that are doubly ruled by curves, provide a grand generalization of most of the previous results. We reconstruct the bound for points and lines, and improve, in certain significant ways, recent bounds involving points and circles (in Sharir, Sheffer and Zahl), and points and arbitrary constant-degree algebraic curves (in Sharir, Sheffer and Solomon). While in these latter instances the bounds are not known (and are strongly suspected not) to be tight, our bounds are, in a certain sense, the best that can be obtained with this approach, given the current state of knowledge. As an application of our point-curve incidence bound, we show that the number of triangles spanned by a set of $n$ points in $R^3$ and similar to a given triangle is $O(n^{15/7})$, which improves the bound of Agarwal et al. Our results are also related to a study by Guth et al.~(work in progress), and have been recently applied in Sharir, Solomon and Zlydenko to related incidence problems in three dimensions.

中文翻译：

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三个维度的曲线发生率

我们研究涉及 $R^3$ 中的点和曲线的关联问题。由 Guth 和 Katz 开创的当前（实际上唯一可行的）解决此类问题的方法需要来自代数几何的各种工具，最显着的是 (i) 多项式划分技术，以及 (ii) 对代数曲面的研究由直线或，在最近的研究中，由某种恒定程度的代数曲线统治。通过利用和改进这些工具，我们在 $R^3$ 中获得了新的和改进的点曲线关联问题的界限。从古斯和卡茨关于点和线的工作开始，之前的几项研究已经考虑了这种发生率。我们的结果基于 Guth 和 Zahl 关于由曲线双重支配的曲面的工作，提供了大部分先前结果的总体概括。我们重建点和线的边界，并以某些重要的方式改进了最近涉及点和圆（在 Sharir、Sheffer 和 Zahl 中）以及点和任意常阶代数曲线（在 Sharir、Sheffer 和 Solomon 中）的边界。虽然在后一种情况下，边界未知（并且强烈怀疑不是严格的），但在某种意义上，鉴于当前的知识状态，我们的边界是通过这种方法可以获得的最好的。作为我们的点曲线关联界的应用，我们表明由 $R^3$ 中的一组 $n$ 点跨越并类似于给定三角形的三角形数量为 $O(n^{15/7} )$，提高了 Agarwal 等人的界限。我们的结果也与 Guth 等人的一项研究有关。~（正在进行中），并且最近已应用于 Sharir，