  One of the earliest results in enumerative combinatorial geometry is the
following theorem of de Bruijn and Erd\H{o}s: Every set of points $E$ in a
projective plane determines at least $|E|$ lines, unless all the points are
contained in a line. Motzkin and others extended the result to higher
dimensions, who showed that every set of points $E$ in a projective space
determines at least $|E|$ hyperplanes, unless all the points are contained in a
hyperplane. Let $E$ be a spanning subset of a $d$-dimensional vector space. We
show that, in the partially ordered set of subspaces spanned by subsets of $E$,
there are at least as many $(d-k)$-dimensional subspaces as there are
$k$-dimensional subspaces, for every $k$ at most $d/2$. This confirms the
"top-heavy" conjecture of Dowling and Wilson for all matroids realizable over
some field. The proof relies on the decomposition theorem package for
$\ell$-adic intersection complexes.
