  A hypertoric variety is a quaternionic analogue of a toric variety. Just as
the topology of toric varieties is closely related to the combinatorics of
polytopes, the topology of hypertoric varieties interacts richly with the
combinatorics of hyperplane arrangements and matroids. Using finite field
methods, we obtain combinatorial descriptions of the Betti numbers of
hypertoric varieties, both for ordinary cohomology in the smooth case and
intersection cohomology in the singular case. We also introduce a conjectural
ring structure on the intersection cohomology of a hypertoric variety.
