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]

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

The realization space of an oriented matroid is the space of all configurations of points ${\displaystyle S\subset {\mathbb {R} }^{n}}$ inducing the same oriented matroid structure on ${\displaystyle S}$.

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

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

${\displaystyle U=U_{1}\coprod \cdots \coprod U_{k}}$ and ${\displaystyle V=V_{1}\coprod \cdots \coprod V_{k}}$

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

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

${\displaystyle U=U_{1}\coprod \cdots \coprod U_{k}}$ and ${\displaystyle V=V_{1}\coprod \cdots \coprod V_{k}}$

with ${\displaystyle U_{i}}$ mapping to ${\displaystyle V_{i}}$ under the natural projection ${\displaystyle \pi }$ deleting the last ${\displaystyle d}$ coordinates. We say that ${\displaystyle \pi :\;U\mapsto V}$ is a stable projection if there exist integer polynomial maps $${\displaystyle \varphi _{1},\ldots ,\varphi _{\ell },\psi _{1},\dots ,\psi _{m}:\;{\mathbb {R} }^{n}\mapsto ({\mathbb {R} }^{d})^{*}}$$ such that $${\displaystyle U_{i}=\{(v,v')\in {\mathbb {R} }^{n+d}\mid v\in V_{i}{\text{ and }}\langle \varphi _{a}(v),v'\rangle >0,\langle \psi _{b}(v),v'\rangle =0{\text{ for }}a=1,\dots ,\ell ,b=1,\dots ,m\}.}$$ The stable equivalence is an equivalence relation on semialgebraic subsets generated by stable projections and rational equivalence.

Theorem (Mnëv's universality theorem):

Let ${\displaystyle V}$ be a semialgebraic subset in ${\displaystyle {\mathbb {R} }^{n}}$ defined over integers. Then ${\displaystyle V}$ is stably equivalent to a realization space of a certain oriented matroid.

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 has been used also by Kapovich and Millson in the study of the moduli spaces of linkages and arrangements.^[3]

• 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.

• Vakil, Ravi (2006), "Murphy's law in algebraic geometry: badly-behaved deformation spaces", Inventiones Mathematicae, 164 (3): 569–590, arXiv:math/0411469, Bibcode:2006InMat.164..569V, doi:10.1007/s00222-005-0481-9, MR 2227692, S2CID 7262537
• Richter-Gebert, Jürgen (1995), "Mnëv's universality theorem revisited", Séminaire Lotharingien de Combinatoire, 34, Article B34h, MR 1399755
• Richter-Gebert, Jürgen (1999), "The universality theorems for oriented matroids and polytopes", in Chazelle, Bernard; Goodman, Jacob E.; Pollack, Richard (eds.), Discrete and Computational Geometry: Ten Years Later, Contemporary Mathematics, vol. 223, Providence, Rhode Island: American Mathematical Society, pp. 269–292, doi:10.1090/conm/223/03144, MR 1661389