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][3]

Oriented matroids[]

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 S induced by hyperplanes in ${\displaystyle {\mathbb {R} }^{n}}$. 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 ${\displaystyle S\subset {\mathbb {R} }^{n}}$ inducing the same oriented matroid structure on S.

Stable equivalence of semialgebraic sets[]

For the purposes of Mnëv'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

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

We say that U and 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,

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

with ${\displaystyle U_{i}}$ mapping to ${\displaystyle V_{i}}$ under the natural projection ${\displaystyle \pi }$ deleting last 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.

Mnëv's universality theorem[]

THEOREM (Mnëv's universality theorem)

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

History[]

Mnëv's universality theorem was discovered by in his 1986 Ph.D. thesis.^[4] 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.

Notes[]

• Universality Theorem, a lecture of Nikolai Mnëv (in Russian).
• Nikolai E. Mnëv, The universality theorems on the classification problem of configuration varieties and convex polytopes varieties (pp. 527–543), in "Topology and geometry: Rohlin Seminar." Edited by O. Ya. Viro. Lecture Notes in Mathematics, 1346. Springer-Verlag, Berlin, 1988.
• 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, S2CID 7262537.
• Richter-Gebert, Jürgen (1995), "Mnëv's Universality Theorem Revisited", Séminaire Lotharingien de Combinatoire, B34h: 15

• Jürgen Richter-Gebert The universality theorems for oriented matroids and polytopes, Contemporary Mathematics 223, 269–292 (1999).

References[]