Incidences between points and lines on two- and three-dimensional varieties

TL;DR

This paper establishes improved bounds on the number of incidences between points and lines on algebraic varieties in four-dimensional space, extending and refining previous results with new bounds for special cases.

Contribution

The authors present new incidence bounds for points and lines on algebraic varieties in higher dimensions, improving previous bounds especially when the degree D is small.

Findings

New incidence bound in $\

Improved bounds for special cases when D is small in $\

Extensions to complex fields with additional terms.

Abstract

Let $P$ be a set of $m$ points and $L$ a set of $n$ lines in $\mathbb R^4$, such that the points of $P$ lie on an algebraic three-dimensional surface of degree $D$ that does not contain hyperplane or quadric components, and no 2-flat contains more than $s$ lines of $L$. We show that the number of incidences between $P$ and $L$ is $$ I(P,L) = O\left(m^{1/2}n^{1/2}D + m^{2/3}n^{1/3}s^{1/3} + nD + m\right) , $$ for some absolute constant of proportionality. This significantly improves the bound of the authors, for arbitrary sets of points and lines in $\mathbb R^4$, when $D$ is not too large. The same bound holds when the three-dimensional surface is embedded in any higher dimensional space. For the proof of this bound, we revisit certain parts of [Sharir-Solomon16], combined with the following new incidence bound. Let $P$ be a set of $m$ points and $L$ a set of $n$ lines in $\mathbb R^d$, for $d\ge 3$, which lie in a common two-dimensional algebraic surface of degree $D$ (assumed to be $\ll n^{1/2}$) that does not contain any 2-flat, so that no 2-flat contains more than $s$ lines of $L$ (here we require that the lines of $L$ also be contained in the surface). Then the number of incidences between $P$ and $L$ is $$ I(P,L) = O\left(m^{1/2}n^{1/2}D^{1/2} + m^{2/3}D^{2/3}s^{1/3} + m + n\right). $$ When $d=3$, this improves the bound of Guth and Katz for this special case, when $D \ll n^{1/2}$. Moreover, the bound does not involve the term $O(nD)$, that arises in most standard approaches, and its removal is a significant aspect of our result. Finally, we also obtain (slightly weaker) variants of both results over the complex field. For two-dimensional varieties, the bound is as in the real case, with an added term of $O(D^3)$. For three-dimensional varieties, the bound is as in the real case, with an added term of $O(D^6)$.