[Paper Review] Incidences between points and lines on a two-dimensional variety

Micha Sharir, Noam Solomon|arXiv (Cornell University)|Jan 13, 2015

Computational Geometry and Mesh Generation24 references4 citations

TL;DR

This paper presents a new, elementary proof of an incidence bound between m points and n lines lying on a two-dimensional algebraic variety of degree D in R^d (d ≥ 3), under the condition that no 2-flat contains more than s ≤ D lines. The key result is an improved bound of O(m^{1/2}n^{1/2}D^{1/2} + m^{2/3}min{n,D^2}^{1/3}s^{1/3} + m + n), which strengthens prior bounds—especially when D is small—by avoiding the O(nD) term common in earlier works.

ABSTRACT

We present a direct and fairly simple proof of the following 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 algebraic two-dimensional surface of degree $D$ that does not contain any 2-flat, so that no 2-flat contains more than $s \le D$ lines of $L$. Then the number of incidences between $P$ and $L$ is $$ I(P,L)=O\left(m^{1/2}n^{1/2}D^{1/2} + m^{2/3}\min\{n,D^{2}\}^{1/3}s^{1/3} + m + n ight). $$ When $d=3$, this improves the bound of Guth and Katz~\cite{GK2} for this special case, when $D$ is not too large. A supplementary feature of this work is a review, with detailed proofs, of several basic (and folklore) properties of ruled surfaces in three dimensions.

Motivation & Objective

• To derive a tighter incidence bound between points and lines lying on a common two-dimensional algebraic variety in R^d for d ≥ 3.
• To eliminate the O(nD) term that appears in prior bounds, such as in Guth and Katz's work, by exploiting the geometric structure of the variety.
• To provide a self-contained, elementary proof that avoids reliance on advanced tools like polynomial partitioning, instead using properties of ruled surfaces.
• To extend the incidence bound to cases where the surface may contain planar components, under bounded line containment per plane.
• To establish a foundation for incidence problems in higher dimensions by proving a bound that holds over both R and C, with potential for generalization.

Proposed method

• Use of algebraic geometry to analyze the structure of a two-dimensional variety V of degree D that does not contain any 2-flat.
• Reduction of the incidence problem to a simpler setting by projecting points and lines onto a generic 2-flat, preserving incidence counts.
• Application of the Szemerédi–Trotter theorem in the planar case as a base case for incidence bounds.
• Decomposition of the line set L into subsets based on their containment in irreducible components of V, particularly focusing on lines in ruled surfaces.
• Bounding incidences involving lines not fully contained in a component by charging them to line-component intersections, which are bounded by nD.
• Replacement of the O(nD) term via a novel argument that leverages the boundedness of s (maximum lines per 2-flat) and the degree D, leading to a tighter O(m^{2/3}min{n,D^2}^{1/3}s^{1/3}) term.

Experimental results

Research questions

• RQ1 Can a tighter incidence bound be derived for points and lines lying on a two-dimensional algebraic variety in R^d (d ≥ 3), without relying on polynomial partitioning?
• RQ2 Is it possible to eliminate the O(nD) term in incidence bounds by exploiting the geometric constraints of the variety and bounded line containment per 2-flat?
• RQ3 How do the incidence bounds change when the containing surface V contains planar components, under the constraint that no 2-flat contains more than s lines?
• RQ4 Can the bound be extended to hold over the complex field, and what are the limitations of current techniques in this context?
• RQ5 What is the tightest possible incidence bound for points and lines on a singly ruled surface in three dimensions, and how does it compare to existing bounds?

Key findings

• The paper establishes a new incidence bound: O(m^{1/2}n^{1/2}D^{1/2} + m^{2/3}min{n,D^2}^{1/3}s^{1/3} + m + n), which improves upon the Guth–Katz bound when D ≪ n^{1/2} in three dimensions.
• The bound avoids the O(nD) term that appears in prior works by using the constraint s ≤ D and the geometric structure of the variety, leading to a significant improvement in the low-degree regime.
• For constant-degree surfaces V that do not contain any 2-flat, the number of incidences is O(m + n), a linear bound that generalizes the planar Szemerédi–Trotter result to higher-dimensional varieties.
• An extension of the bound is provided for surfaces that contain planar components, yielding O(m^{2/3}s^{2/3} + m + n) incidences when no 2-flat contains more than s lines.
• The proof is largely elementary and self-contained, avoiding the need for polynomial partitioning and relying instead on algebraic geometry and ruled surface properties.
• The result holds over both R and C, though the final step using the Guth–Katz bound is currently restricted to R; the authors suggest that a complex extension would require a different approach to replace this step.