Acta Mathematica

Volume 218 (2017)

Number 2

Enumeration of points, lines, planes, etc.

Pages: 297 – 317

DOI: http://dx.doi.org/10.4310/ACTA.2017.v218.n2.a2

Authors

June Huh (Institute for Advanced Study, Princeton, New Jersey, U.S.A.; and Korea Institute for Advanced Study, Seoul, Korea)

Botong Wang (University of Wisconsin, Madison, Wisc., U.S.A.)

Abstract

One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erdős: Every set of points $E$ in a projective plane determines at least $\lvert E \rvert$ lines, unless all the points are contained in a line. The result was extended to higher dimensions by Motzkin and others, who showed that every set of points $E$ in a projective space determines at least $\lvert E \rvert$ hyperplanes, unless all the points are contained in a hyperplane. Let $E$ be a spanning subset of an $r$-dimensional vector space. We show that, in the partially ordered set of subspaces spanned by subsets of $E$, there are at least as many $(r-p)$-dimensional subspaces as there are $p$-dimensional subspaces, for every $p$ at most $\frac{1}{2} r$. This confirms the “top-heavy” conjecture by Dowling and Wilson for all matroids realizable over some field. The proof relies on the decomposition theorem package for $\ell$-adic intersection complexes.

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June Huh was supported by a Clay Research Fellowship and NSF Grant DMS-1128155.

Received 9 October 2016

Received revised 30 January 2017

Published 27 November 2017