Incidences with curves in three dimensions

Abstract

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 [23,24], 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 [25], by algebraic curves of some constant degree. By exploiting and refining these tools, we obtain new and improved bounds for pointcurve 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 [24]. Our results, which are based on the work of Guth and Zahl [25] 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 signifcant ways, recent bounds involving points and circles (in [36]), and points and arbitrary constant-degree algebraic curves (in [35]). 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. [1]. Our results are also related to a study by Guth et al. (work in progress), and have been recently applied in Sharir et al. [42] to related incidence problems in three dimensions.