Ordinary lines, point degrees, and matroid density

Abstract

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 rank-$r$ complex-representable matroid excluding $U_{2,\ell}$ as a minor, which removes the sufficiently-large-rank assumption from the bound of Geelen, Nelson, and Walsh. An arithmetic application shows that rank-$r$ integer matrices whose maximal minors have absolute value at most $Δ$ have at most $O(Δr^2)$ distinct columns, resolving a question of Averkov and Schymura. We prove that a set of $n$ points in $\mathbb{C}^3$ determines at least $n^2/40$ ordinary lines whenever no plane contains more than $2n/3$ of the points. This answers a question of Basit, Dvir, Saraf, and Wolf. We also prove upper bounds on average line length and lower bounds on the number of points incident to many spanned lines in high-dimensional point sets. The proofs combine contraction of concentrated flats with normalization of the point vectors and second- and fourth-moment estimates on their pairwise correlations. This approach builds on the design matrix method.