On four-rich points defined by pencils

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-Szabó 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-Szabó 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 ${\mathbb{C}}^2$.