Mnev's universality theorem

Oriented matroids

For the purposes of Mnev's universality, an oriented matroid of a finite subset S ⊂ R n is a list of all partitions of points in S induced by hyperplanes in R n . In particular, the structure of oriented matroid contains full information on the incidence relations in S, inducing on S a matroid structure.

The realization space of an oriented matroid is the space of all configurations of points S ⊂ R n inducing the same oriented matroid structure on S.

Stable equivalence of semialgebraic sets

For the purposes of Mnev's Universality, the stable equivalence of semialgebraic sets is defined as follows.

Let U, V be semialgebraic sets, obtained as a disconnected union of connected semialgebraic sets

We say that U and V are rationally equivalent if there exist homeomorphisms U i ↦ ϕ i V i defined by rational maps.

Let U ⊂ R n + d , V ⊂ R n be semialgebraic sets,

with U i mapping to V i under the natural projection π deleting last d coordinates. We say that π : U ↦ V is a stable projection if there exist integer polynomial maps

such that

The stable equivalence is an equivalence relation on semialgebraic subsets generated by stable projections and rational equivalence.

Mnev's Universality theorem

THEOREM (Mnev's universality theorem)
Let V be a semialgebraic subset in R n defined over integers. Then V is stably equivalent to a realization space of a certain oriented matroid.

History

Mnev's universality theorem was discovered by Nikolai Mnev in his 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.