Library Header Image
LSE Research Online LSE Library Services

On sets defining few ordinary hyperplanes

Lin, Aaron and Swanepoel, Konrad ORCID: 0000-0002-1668-887X (2020) On sets defining few ordinary hyperplanes. Discrete Analysis. ISSN 2397-3129

[img] Text (On sets defining few ordinary hyperplanes) - Published Version
Available under License Creative Commons Attribution.

Download (371kB)

Identification Number: 10.19086/da.11949


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 Pis a hyperplane containing exactly d points of P. We show that if d > 4, the numberof ordinary hyperplanes of P is at least􀀀n􀀀1d􀀀1􀀀 Od(nb(d􀀀1)=2c) if n is sucientlylarge depending on d. This bound is tight, and given d, we can calculate the exactminimum number for suciently large n. This is a consequence of a structuretheorem for sets with few ordinary hyperplanes: For any d > 4 and K > 0, ifn > CdK8 for some constant Cd > 0 depending on d, and P spans at most K􀀀n􀀀1d􀀀1ordinary hyperplanes, then all but at most Od(K) points of P lie on a hyperplane,an elliptic normal curve, or a rational acnodal curve. We also nd the maximumnumber of (d+1)-point hyperplanes, solving a d-dimensional analogue of the orchardproblem. Our proofs rely on Green and Tao's results on ordinary lines, our earlierwork on the 3-dimensional case, as well as results from classical algebraic geometry.

Item Type: Article
Official URL:
Additional Information: © 2020 The Authors
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 10 Feb 2020 10:18
Last Modified: 01 Aug 2020 23:39

Actions (login required)

View Item View Item


Downloads per month over past year

View more statistics