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 $d\ge 4$, the number of ordinary hyperplanes of $P$ is at least $\left(\genfrac{}{}{0px}{}{n-1}{d-1}\right)-O_d(n^{\lfloor (d-1)\mathrm{/}2\rfloor })$ if $n$ 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 $d\ge 4$ and $K>0$, if $n\ge C_d{K}^8$ for some constant $C_d>0$ depending on $d$ and $P$ spans at most $K\left(\genfrac{}{}{0px}{}{n-1}{d-1}\right)$ 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.