Spanned lines and Langer's inequality

TL;DR

This paper explores how Langer's inequality from algebraic geometry provides improved bounds in combinatorial geometry, specifically in incidences, Beck's theorem, and related problems, by consolidating known results and discussing potential enhancements.

Contribution

It demonstrates that Langer's inequality can be used to derive better constants in combinatorial geometry theorems like Beck's theorem, consolidating existing proofs and results.

Findings

Improved constants in Beck's theorem using Langer's inequality

Lower bounds on incidences between points and lines

Discussion of potential further improvements

Abstract

We collect some results in combinatorial geometry that follow from an inequality of Langer in algebraic geometry. Langer's inequality gives a lower bound on the number of incidences between a point set and its spanned lines, and was recently used by Han to improve the constant in the weak Dirac conjecture. Here we observe that this inequality also leads to improved constants in Beck's theorem, which states that a finite point set in the real or complex plane has many points on a line or spans many lines. Most of the proofs that we use are not original, and the goal of this note is mainly to carefully record the quantitative results in one place. We also include some discussion of possible further improvements to these statements.