Mnëv's universality theorem explained

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

be a semialgebraic variety in defined over the integers. Then 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

See main article: Oriented matroid.

For the purposes of this article, an oriented matroid of a finite subset

is the list of partitions of 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 , inducing on a matroid structure.

The realization space of an oriented matroid is the space of all configurations of points

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

and be semialgebraic sets, obtained as a disjoint union of connected semialgebraic sets and

We say that

and are rationally equivalent if there exist homeomorphisms defined by rational maps.

Let

be semialgebraic sets, and with mapping to under the natural projection deleting the last coordinates. We say that is a stable projection if there exist integer polynomial maps$$\backslash varphi\_1,\; \backslash ldots,\; \backslash varphi\_\backslash ell,\; \backslash psi\_1,\; \backslash dots,\; \backslash psi\_m:\backslash ;\; ^n\; \backslash to\; ^d$$such that$$U\_i\; =\backslash .$$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.

Further reading

• Convex Polytopes by Branko Grünbaum, revised edition, a book that includes material on the theorem and its relation to the realizability of polytopes from their combinatorial structures.