Ordinary lines, point degrees, and matroid density

For every set of $n$ points in $\mathbb{C}^d$ with full affine span, we prove that some point lies on at least $n+1-2n/(d+1)$ spanned lines, which is best possible in every dimension. By successive contraction, this implies the sharp bound $r+(\ell-3)\binom r2$ on the number of elements of a simple