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.
[1][2] 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: The theorem was discovered by Nikolai Mnëv in his 1986 Ph.D. thesis.
For the purposes of this article, an oriented matroid of a finite subset
induced by hyperplanes in
(each oriented hyperplane partitions
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
The realization space of an oriented matroid is the space of all configurations of points
inducing the same oriented matroid structure.
For the purpose of this article stable equivalence of semialgebraic sets is defined as described below.
be semialgebraic sets, obtained as a disjoint union of connected semialgebraic sets We say that
are rationally equivalent if there exist homeomorphisms
defined by rational maps.
under the natural projection
is a stable projection if there exist integer polynomial maps
The stable equivalence is an equivalence relation on semialgebraic subsets generated by stable projections and rational equivalence.
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.
[3] 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.