Incidences with curves in three dimensions

TL;DR

This paper advances the understanding of point-curve incidences in three-dimensional space using algebraic geometry tools, providing new bounds and applications to geometric problems like counting specific triangles.

Contribution

It refines algebraic geometry techniques to establish improved bounds for point-curve incidences in 3D, generalizing previous results and applying them to geometric counting problems.

Findings

New bounds for point-curve incidences in R^3

Improved triangle counting bound of O(n^{15/7})

Generalization of previous incidence bounds to algebraic curves

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, 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.