A note on rich lines in truly high dimensional sets

TL;DR

This paper extends geometric combinatorics results to high-dimensional complex spaces, showing that many rich lines imply a large subset lying in a hyperplane, with implications for understanding point-line incidences.

Contribution

It refines existing arguments to demonstrate that abundant rich lines in high dimensions are concentrated in a hyperplane, advancing the understanding of geometric configurations.

Findings

Many rich lines in high-dimensional sets lie in a common hyperplane

The argument adapts and modifies previous combinatorial geometric methods

Results relate to and extend prior work by Dvir and Gopi

Abstract

We modify an argument of Hablicsek and Scherr to show that if a collection of points in $\mathbb{C}^d$ spans many $r$--rich lines, then many of these lines must lie in a common $(d-1)$--flat. This is closely related to a previous result of Dvir and Gopi.