On sets defining few ordinary hyperplanes

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 ordinary hyperplanes of P is at least 􀀀n􀀀1 d􀀀1 􀀀 Od(nb(d􀀀1)=2c) if n is suciently large depending on d. This bound is tight, and given d, we can calculate the exact minimum number for suciently 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 > CdK8 for some constant Cd > 0 depending on d, and P spans at most K 􀀀n􀀀1 d􀀀1 ordinary 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 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.