Geometry Seminar

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Web geometry and orchard problem

Speaker: Mehdi Makhul, LSE

Location: Online Zoom-only

Videoconference link: https://youtu.be/UrFl196kR4E

Date: Tuesday, May 6, 2025, 2 p.m.

Synopsis:

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 hold: the degree of \(C\) must be \(d+ 1\); and the curve \(C\) is the complete intersection of \(\binom{d}{2}-1\) quadric hypersurfaces. Our approach relies on the theory of web geometry and the Elekes–Szabó Theorem-a cornerstone of incidence geometry-both of which provide the structural basis for our analysis.

Notes:

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