Web geometry and the orchard problem

Abstract: Let $P$ be a set of $n$ points in the plane, not all lying on a single line. The orchard planting problem asks for the maximum number of lines passing through exactly three points of $P$. Green and Tao showed that the maximum possible number of such lines for an $n$-element set is~$\lfloor \frac{n(n-3)}{6} \rfloor+1$. Lin and Swanepoel also investigated a generalization of the orchard problem in higher dimensions. Specifically, if $P$ is a set of $n$ points in $d$-dimensional space, they established an upper bound for the maximum number of hyperplanes passing through exactly $d+1$ points of $P$. Our goal is to describe the structural properties of configurations that achieve near-optimality in the asymptotic regime. Let $C \subset \mathbb{R}^d$ be an algebraic curve of degree~$r$, and suppose that $P \subset C$ is a set of $n$ points. If $P$ determines at least~$cn^d$ hyperplanes, each passing through exactly $d+1$ points of $P$, then the following must hold: The degree of $C$ must be $d+1$; and the curve $C$ is the complete intersection of ${d\choose 2}-1$ quadric hypersurfaces. Our approach relies on the theory of web geometry and the Elekes-Szab\'o Theorem-a cornerstone of incidence geometry-both of which provide the structural basis for our analysis. Joint work with Konrad Swanepoel.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)