Untitled

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 show that if d 4, the number of ordinaryif n is sufficiently large depending on d. This bound is tight, and given d, we can calculate the exact minimum number for sufficiently large n. This is a consequence of a structure theorem for sets with few ordinary hyperplanes: For any d 4 and K > 0, if n C d K 8 for some constant C d > 0 depending on d, and P spans at most K n−1 d−1 ordinary hyperplanes, then all but at most O d (K) points of P lie on a hyperplane, an elliptic normal curve, or a rational acnodal curve. We also find the maximum number of (d + 1)-point hyperplanes, solving a d-dimensional analogue of the orchard problem. Our proofs rely on Green and Tao's results on ordinary lines, our earlier work on the 3-dimensional case, as well as results from classical algebraic geometry.