Mnëv's universality theorem
Introduction In algebraic geometry, '''Mnëv's universality theorem''' is a result which can be used to represent algebraic (or semi algebraic) varieties as realizations of oriented matroids, a notion of combinatorics.
Oriented matroids For the purposes of Mnëv's universality, an oriented matroid of a finite subset  is a list of all partitions of points in  induced by hyperplanes in . In particular, the structure of oriented matroid contains full information on 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 on .
Stable equivalence of semialgebraic sets For the purposes of universality, the '''stable equivalence''' of semialgebraic sets is defined as follows.Let  and  be semialgebraic sets, obtained as a disconnected union of connected semialgebraic setsWe say that  and  are ''rationally equivalent'' if there exist homeomorphisms  defined by rational maps.Let  be semialgebraic sets,with  mapping to  under the natural projection  deleting the last  coordinates. We say that  is a ''stable projection'' if there exist integer polynomial mapssuch thatThe ''stable equivalence'' is an equivalence relation on semialgebraic subsets generated by stable projections and rational equivalence.
Mnëv's universality theorem Theorem (''Mnëv's universality theorem''):Let  be a semialgebraic subset in  defined over integers. Then  is stably equivalent to a realization space of a certain oriented matroid.
History Mnëv's universality theorem was discovered by Nikolai Mnëv in his 1986 Ph.D. thesis. It 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 have been used also by Kapovich and Millson in the study of the moduli spaces of linkages and arrangements.
See also * Convex Polytopes a book that includes material on the theorem and its relation to the realizability of polytopes from their combinatorial structures.
References
Further reading ***