On four-rich points defined by pencils

TL;DR

This paper establishes upper bounds on the number of four-rich points formed by algebraic objects like sheaves of planes and spheres, using advanced combinatorial geometry theorems.

Contribution

It introduces new bounds on four-rich points for algebraic configurations in three and two dimensions using the Elekes-Szabó theorem.

Findings

Bound of O(n^{8/3}) four-rich points for four sheaves of planes in P^3

Upper bounds on four-rich points for configurations of spheres in C^3

Application of Elekes-Szabó theorem to various geometric configurations

Abstract

In this paper we study the number of four-rich points defined by pencils of certain algebraic objects. Our main result concerns the number of four-rich points defined by four sheaves of planes; under certain non-degeneracy conditions, we prove that four sheaves of $n$ planes in $\mathbb P^3$ determine at most $O(n^{8/3})$ four-rich points. We prove this using the four dimensional Elekes-Szab\'{o} theorem. Using the same method, we prove an upper bound on the number of four-rich points determined by four sets of concentric spheres in $\mathbb C^3$. Furthermore, using the same technique with the 3-d Elekes-Szab\'{o} theorem, one can prove upper bounds on four-rich points determined by various configurations of lines/circles in the plane $\mathbb C^2$; we give one such example, involving two pencils of lines and two pencils of concentric circles in $\mathbb C^2$.