Ordinary hyperspheres and spherical curves

An ordinary hypersphere of a set of points in real d-space, where no d + 1 points lie on a (d - 2)-sphere or a (d - 2)-flat, is a hypersphere (including the degenerate case of a hyperplane) that contains exactly d + 1 points of the set. Similarly, a (d + 2)-point hypersphere of such a set is one that contains exactly d + 2 points of the set. We find the minimum number of ordinary hyperspheres, solving the d-dimensional spherical analogue of the Dirac-Motzkin conjecture for d ≥ 3. We also find the maximum number of (d + 2)-point hyperspheres in even dimensions, solving the d-dimensional spherical analogue of the orchard problem for even d ≥ 4

On sets defining few ordinary circles

An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least 1/4n2 − O(n) ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most 1/24n 3−O(n2) circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most Kn2 ordinary circles, , then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves

Structure theorems and extremal problems in incidence geometry

In this thesis, we prove variants and generalisations of the Sylvester-Gallai theorem, which states that a finite non-collinear point set in the plane spans an ordinary line. Green and Tao proved a structure theorem for sufficiently large sets spanning few ordinary lines, and used it to find exact extremal numbers for ordinary and 3-rich lines, solving the Dirac-Motzkin conjecture and the classical orchard problem respectively. We prove structure theorems for sufficiently large sets spanning few ordinary planes, hyperplanes, circles, and hyperspheres, showing that such sets lie mostly on algebraic curves (or on a hyperplane or hypersphere). We then use these structure theorems to solve the corresponding analogues of the Dirac-Motzkin conjecture and the orchard problem. For planes in 3-space and circles in the plane, we are able to find exact extremal numbers for ordinary and 4-rich planes and circles. We also show that there are irreducible rational space quartics such that any n-point subset spans only O(n8=3) coplanar quadruples, answering a question of Raz, Sharir, and De Zeeuw [51]. For hyperplanes in d-space, we are able to find tight asymptotic bounds on the extremal numbers for ordinary and (d + 1)-rich hyperplanes. This also gives a recursive method to compute exact extremal numbers for a fixed dimension d. For hyperspheres in d-space, we are able to find a tight asymptotic bound on the minimum number of ordinary hyperspheres, and an asymptotic bound on the maximum number of (d + 2)-rich hyperspheres that is tight in even dimensions. The recursive method in the hyperplanes case also applies here. Our methods rely on Green and Tao's results on ordinary lines, as well as results from classical algebraic geometry, in particular on projections, inversions, and algebraic curves