Konrad Swanepoel

Ordinary hyperplanes and space curves

It was shown by Green and Tao (2013) that if n is large enough, a set of n points in the plane, not all on a line, determine at least n/2 ordinary lines, that is, lines that pass through exactly two points of the set. They also characterised the sets with O(n) ordinary lines: almost all their points are contained in a cubic curve. Simeon Ball (2018) solved an analogous problem in 3-space: determine the smallest number of ordinary planes in a set of n points, not all on a plane, and with no three points on a line, where an ordinary plane intersects the set in exactly three points. He showed that if such a set has only O(n^2) ordinary planes, almost all the points lie on the intersection of two quadric surfaces. In this talk I discuss this result, indicate an alternative proof, and consider further generalisations to higher dimensional space: extremal sets lie on special space curves. The proofs rely on results from elementary classical algebraic geometry and invariant theory. This is joint work with Aaron Lin.

Konrad Swanepoel is an assistant professor in the department of mathematics at the London School of Economics. He studies combinatorial and discrete geometry, convex geometry, geometric networks, and the geometry of finite-dimensional normed spaces