Ordinary planes, coplanar quadruples, and space quartics

TL;DR

This paper characterizes finite point sets in 3D space with few ordinary planes, extending previous results and providing bounds on coplanar quadruples on space quartic curves, combining combinatorial geometry and algebraic curve theory.

Contribution

It introduces a structure theorem for point sets with few ordinary planes and extends bounds on coplanar quadruples on space quartic curves, offering an alternative approach to existing results.

Findings

Established a structure theorem for sets with few ordinary planes.

Provided bounds on coplanar quadruples on rational space quartic curves.

Extended and unified previous results on ordinary lines and planes.

Abstract

An ordinary plane of a finite set of points in real 3-space with no three collinear is a plane intersecting the set in exactly three points. We prove a structure theorem for sets of points spanning few ordinary planes. Our proof relies on Green and Tao's work on ordinary lines in the plane, combined with classical results on space quartic curves and non-generic projections of curves. This gives an alternative approach to Ball's recent results on ordinary planes, as well as extending them. We also give bounds on the number of coplanar quadruples determined by a finite set of points on a rational space quartic curve in complex 3-space, answering a question of Raz, Sharir and De Zeeuw [Israel J. Math. 227 (2018)].