Mnëv's universality theorem

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.^[1]^[2]

Oriented matroids

For the purposes of Mnëv's universality, an oriented matroid of a finite subset $S\subset {\mathbb {R} }^{n}$ is a list of all partitions of points in $S$ induced by hyperplanes in ${\mathbb {R} }^{n}$ . 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 $S\subset {\mathbb {R} }^{n}$ inducing the same oriented matroid structure on $S$ .

Stable equivalence of semialgebraic sets

For the purposes of universality, the stable equivalence of semialgebraic sets is defined as follows.

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

U=U_{1}\coprod \cdots \coprod U_{k}

and

V=V_{1}\coprod \cdots \coprod V_{k}

We say that $U$ and $V$ are rationally equivalent if there exist homeomorphisms $U_{i}{\stackrel {\varphi _{i}}{\mapsto }}V_{i}$ defined by rational maps.

Let $U\subset {\mathbb {R} }^{n+d},V\subset {\mathbb {R} }^{n}$ be semialgebraic sets,

U=U_{1}\coprod \cdots \coprod U_{k}

and

V=V_{1}\coprod \cdots \coprod V_{k}

with $U_{i}$ mapping to $V_{i}$ under the natural projection $\pi$ deleting the last $d$ coordinates. We say that $\pi :\;U\mapsto V$ is a stable projection if there exist integer polynomial maps

\varphi _{1},\ldots ,\varphi _{\ell },\psi _{1},\dots ,\psi _{m}:\;{\mathbb {R} }^{n}\mapsto ({\mathbb {R} }^{d})^{*}

such that

U_{i}=\{(v,v')\in {\mathbb {R} }^{n+d}\mid v\in V_{i}{\text{ and }}\langle \varphi _{a}(v),v'\rangle >0,\langle \psi _{b}(v),v'\rangle =0{\text{ for }}a=1,\dots ,\ell ,b=1,\dots ,m\}.

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