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Space quartics, ordinary planes and coplanar quadruples

Banff International Research Station for Mathematical Innovation and Discovery

It was shown by Raz-Sharir-De Zeeuw (2016) that the number of coplanar quadruples among n points on an algebraic curve in complex 3-space not containing a planar component or a component of degree 4, is O(n^{8/3}). We complement their result by characterizing the degree 4 space curves in which n points on the curve always have a subcubic number of coplanar quadruples. This also gives a characterization of the plane curves of degree 3 and 4 in which n points on the curve always have a subcubic number of concyclic quadruples. We use the 4-dimensional Elekes-Szabo theorem of Raz-Sharir-De Zeeuw and some old results from classical invariant theory. Simeon Ball (2016) showed that a set spanning real 3-space, no 3 collinear, with only Kn^2 ordinary planes, lies on the intersection of two quadrics, up to O(K) points. His proof is based on results of Green and Tao, and also generalizes their proof to 3-space. We find a significant simplification of his proof that avoids 3-dimensional dual configurations, using Bezout's theorem and the above-mentioned results from classical invariant theory. This is joint work with Aaron Lin.

Mathematics; Convex and discrete geometry; Combinatorics

video/mp4

Author affiliation: London School of Economics and Political Science

2018-08-08

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/66695

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/