Aaron Lin
London School of Economics and Political Science

Tuesday, May 15 at 2:30 PM

Abstract

Let $P$ be a set of $n$ points in real projective $d$-space, not all contained in a hyperplane, such that any $d$ points span a hyperplane. An ordinary hyperplane of $P$ is a hyperplane containing exactly $d$ points of $P$. We prove a structure theorem for sets with few ordinary hyperplanes for each $d \geqslant 4$: If $n$ is sufficiently large depending on $K$, and $P$ spans at most $Kn^{d-1}$ ordinary hyperplanes, then all but $O(K)$ points of $P$ lie on a hyperplane, a genus 1 normal curve, or a rational acnodal curve. We also find the exact minimum number of ordinary hyperplanes for all sufficiently large $n$. The case $d = 3$ was first proved by Ball (we have an independent proof), based on Green and Tao’s work on the $d = 2$ case. Our proofs also rely on Green and Tao’s results, as well as results from classical algebraic geometry. This is joint work with Konrad Swanepoel.

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