On rich points and incidences with restricted sets of lines in 3-space

TL;DR

This paper establishes new bounds on incidences and rich points for sets of lines in 3D space constrained by algebraic varieties, with applications to distance problems and incidences in higher dimensions.

Contribution

It introduces bounds on rich points and incidences for lines in 3-space constrained by algebraic varieties, extending incidence theory with algebraic geometry tools.

Findings

Bound on the number of r-rich points for 2D varieties in R^3

Incidence bounds between lines and points on algebraic varieties in R^3

Incidence bounds for lines on quadratic hypersurfaces in R^4

Abstract

Let $L$ be a set of $n$ lines in $R^3$ that is contained, when represented as points in the four-dimensional Pl\"ucker space of lines in $R^3$, in an irreducible variety $T$ of constant degree which is \emph{non-degenerate} with respect to $L$ (see below). We show: \medskip \noindent{\bf (1)} If $T$ is two-dimensional, the number of $r$-rich points (points incident to at least $r$ lines of $L$) is $O(n^{4/3+\epsilon}/r^2)$, for $r \ge 3$ and for any $\epsilon>0$, and, if at most $n^{1/3}$ lines of $L$ lie on any common regulus, there are at most $O(n^{4/3+\epsilon})$ $2$-rich points. For $r$ larger than some sufficiently large constant, the number of $r$-rich points is also $O(n/r)$. As an application, we deduce (with an $\epsilon$-loss in the exponent) the bound obtained by Pach and de Zeeuw (2107) on the number of distinct distances determined by $n$ points on an irreducible algebraic curve of constant degree in the plane that is not a line nor a circle. \medskip \noindent{\bf (2)} If $T$ is two-dimensional, the number of incidences between $L$ and a set of $m$ points in $R^3$ is $O(m+n)$. \medskip \noindent{\bf (3)} If $T$ is three-dimensional and nonlinear, the number of incidences between $L$ and a set of $m$ points in $R^3$ is $O\left(m^{3/5}n^{3/5} + (m^{11/15}n^{2/5} + m^{1/3}n^{2/3})s^{1/3} + m + n \right)$, provided that no plane contains more than $s$ of the points. When $s = O(\min\{n^{3/5}/m^{2/5}, m^{1/2}\})$, the bound becomes $O(m^{3/5}n^{3/5}+m+n)$. As an application, we prove that the number of incidences between $m$ points and $n$ lines in $R^4$ contained in a quadratic hypersurface (which does not contain a hyperplane) is $O(m^{3/5}n^{3/5} + m + n)$. The proofs use, in addition to various tools from algebraic geometry, recent bounds on the number of incidences between points and algebraic curves in the plane.