Realization of semialgebraic sets by points
In mathematics, Mnëv's universality theorem is a result in the intersection of combinatorics and algebraic geometry used to represent algebraic (or semialgebraic) varieties as realization spaces of oriented matroids.
Informally it can also be understood as the statement that point configurations of a fixed combinatorics can show arbitrarily complicated behavior.
The precise statement is as follows:
 Let formula_1 be a semialgebraic variety in formula_2 defined over the integers. Then formula_1 is stably equivalent to the realization space of some oriented matroid.
The theorem was discovered by Nikolai Mnëv in his 1986 Ph.D. thesis. 
Oriented matroids.
For the purposes of this article, an "oriented matroid" of a finite subset formula_4 is the list of partitions of formula_5 induced by hyperplanes in formula_2 (each oriented hyperplane partitions formula_5 into the points on the "positive side" of the hyperplane, the points on the "negative side" of the hyperplane, and the points that lie on the hyperplane). In particular, an oriented matroid contains the full information of the incidence relations in formula_5, inducing on formula_5 a matroid structure.
The realization space of an oriented matroid is the space of all configurations of points formula_4 inducing the same oriented matroid structure.
Stable equivalence of semialgebraic sets.
For the purpose of this article stable equivalence of semialgebraic sets is defined as described below.
Let formula_11 and formula_1 be semialgebraic sets, obtained as a disjoint union of connected semialgebraic sets
formula_13 and formula_14
We say that formula_11 and formula_1 are "rationally equivalent" if there exist homeomorphisms formula_17 defined by rational maps.
Let formula_18 be semialgebraic sets,
formula_13 and formula_14
with formula_21 mapping to formula_22 under the natural projection formula_23 deleting the last formula_24 coordinates. We say that formula_25 is a "stable projection" if there exist integer polynomial maps
formula_26
such that
formula_27
The "stable equivalence" is an equivalence relation on semialgebraic subsets generated by stable projections and rational equivalence.
Implications.
Mnëv's universality theorem has numerous applications in algebraic geometry, due to Laurent Lafforgue, Ravi Vakil and others, allowing one to construct moduli spaces with arbitrarily bad behaviour. This theorem together with Kempe's universality theorem has been used also by Kapovich and Millson in the study of the moduli spaces of linkages and arrangements.
Mnëv's universality theorem also gives rise to the "universality theorem for convex polytopes". In this form it states that every semialgebraic set is stably equivalent to the realization space of some convex polytope. Jürgen Richter-Gebert showed that universality already applies to polytopes of dimension four.
References.
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