A short proof of Rudnev's point-plane incidence bound

TL;DR

This paper presents a simplified geometric proof of Rudnev's point-plane incidence bound over arbitrary fields, replacing complex mappings with a direct approach to reduce technical complexity.

Contribution

It introduces a straightforward geometric map that directly relates point-plane incidences to line-line intersections, simplifying Rudnev's original proof.

Findings

Provides a shorter, more accessible proof of Rudnev's theorem

Reduces technical complexity in incidence geometry proofs

Demonstrates the effectiveness of direct geometric mappings

Abstract

In this note we give a shortened proof of a theorem of Rudnev, which bounds the number of incidences between points and planes over an arbitrary field. Rudnev's proof uses a map that goes via the four-dimensional Klein quadric to a three-dimensional space, where it applies a bound of Guth and Katz on intersection points of lines. We describe a simple geometric map that directly sends point-plane incidences to line-line intersections in space, allowing us to reprove Rudnev's theorem with fewer technicalities.