Mnev's universality theorem claims that any semialgebraic set is the realization space of some oriented matroid. Moreover, the rank of the or matroid can be prescribed in advance.
1.-Are there interesting corollaries to Mnev's theorem? I am aware of interesting algorithmic consequences.
Geometric consequences? Examples in which the theorem is used to prove that other moduli spaces can also be wild?
MacPherson's definition of "combinatorial differentiable manifolds" and oriented matroid bundles are based on a local system of oriented matroids over a simplicial complex. Is there some implication from Mnev's theorem to the theory of combinatorial differentiable manifolds.
What about proofs that would be easy (or statemens that would be true) if realization spaces of oriented matroids where better behaved, say connected, or contractible..
2.-Are there quantitative versions of this theorem relating (say) the number and degrees of the defining polynomial (in)equalities or the betty numbers of the semialgebraic set with the rank and number of elements in the corresponding or-mat.