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Ordinary planes, coplanar quadruples, and space quartics

Lin, A and Swanepoel, Konrad ORCID: 0000-0002-1668-887X (2019) Ordinary planes, coplanar quadruples, and space quartics. Journal of the London Mathematical Society, 100 (3). pp. 937-956. ISSN 0024-6107

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Identification Number: 10.1112/jlms.12251


An ordinary plane of a finite set of points in real 3-space with no three collinear is a plane intersecting the set in exactly three points. We prove a structure theorem for sets of points spanning few ordinary planes. Our proof relies on Green and Tao's work on ordinary lines in the plane, combined with classical results on space quartic curves and non-generic projections of curves. This gives an alternative approach to Ball's recent results on ordinary planes, as well as extending them. We also give bounds on the number of coplanar quadruples determined by a finite set of points on a rational space quartic curve in complex 3-space, answering a question of Raz, Sharir, and De Zeeuw [Israel J. Math. 227 (2018) 663–690].

Item Type: Article
Additional Information: © 2019 2019 London Mathematical Society
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 29 Apr 2019 11:00
Last Modified: 16 May 2024 02:42

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