Enumeration of points, lines, planes, etc.
-- June Huh, November 11, 2016
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One of the earliest results in enumerative combinatorial geometry is
the following theorem of de Bruijn and Erd&#337;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 poset 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
from 1974 for all matroids realizable over some field. The proof
relies on the decomposition theorem package for l-adic intersection
complexes. Joint work with Botong Wang.