Mnëv's universality theorem

Short description: Realization of semialgebraic sets by points

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 [math]\displaystyle{ S\subset {\mathbb R}^n }[/math] is a list of all partitions of points in [math]\displaystyle{ S }[/math] induced by hyperplanes in [math]\displaystyle{ {\mathbb R}^n }[/math]. In particular, the structure of oriented matroid contains full information on the incidence relations in [math]\displaystyle{ S }[/math], inducing on [math]\displaystyle{ S }[/math] a matroid structure.

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

Stable equivalence of semialgebraic sets

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

Let [math]\displaystyle{ U }[/math] and [math]\displaystyle{ V }[/math] be semialgebraic sets, obtained as a disconnected union of connected semialgebraic sets

[math]\displaystyle{ U=U_1\coprod \cdots\coprod U_k }[/math] and [math]\displaystyle{ V=V_1\coprod \cdots\coprod V_k }[/math]

We say that [math]\displaystyle{ U }[/math] and [math]\displaystyle{ V }[/math] are rationally equivalent if there exist homeomorphisms [math]\displaystyle{ U_i \stackrel {\varphi_i} \mapsto V_i }[/math] defined by rational maps.

Let [math]\displaystyle{ U\subset {\mathbb R}^{n+d}, V\subset {\mathbb R}^{n} }[/math] be semialgebraic sets,

[math]\displaystyle{ U=U_1\coprod \cdots\coprod U_k }[/math] and [math]\displaystyle{ V=V_1\coprod \cdots\coprod V_k }[/math]

with [math]\displaystyle{ U_i }[/math] mapping to [math]\displaystyle{ V_i }[/math] under the natural projection [math]\displaystyle{ \pi }[/math] deleting the last [math]\displaystyle{ d }[/math] coordinates. We say that [math]\displaystyle{ \pi:\; U \mapsto V }[/math] is a stable projection if there exist integer polynomial maps [math]\displaystyle{ \varphi_1, \ldots, \varphi_\ell, \psi_1, \dots, \psi_m:\; {\mathbb R}^n \mapsto ({\mathbb R}^d)^* }[/math] such that [math]\displaystyle{ U_i =\{ (v,v') \in {\mathbb R}^{n+d}\mid v\in V_i \text{ and } \langle \varphi_a(v), v'\rangle \gt 0, \langle \psi_b(v), v'\rangle=0 \text{ for } a=1,\dots, \ell, b = 1, \dots, m\}. }[/math] 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 [math]\displaystyle{ V }[/math] be a semialgebraic subset in [math]\displaystyle{ {\mathbb R}^n }[/math] defined over integers. Then [math]\displaystyle{ V }[/math] 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.

Further reading

• Vakil, Ravi (2006), "Murphy's law in algebraic geometry: badly-behaved deformation spaces", Inventiones Mathematicae 164 (3): 569–590, doi:10.1007/s00222-005-0481-9, Bibcode: 2006InMat.164..569V 
• Richter-Gebert, Jürgen (1995), "Mnëv's universality theorem revisited", Séminaire Lotharingien de Combinatoire 34: Article B34h, https://www.emis.de/journals/SLC/wpapers/s34berlin.html 
• Richter-Gebert, Jürgen (1999), "The universality theorems for oriented matroids and polytopes", in Chazelle, Bernard; Goodman, Jacob E.; Pollack, Richard, Discrete and Computational Geometry: Ten Years Later, Contemporary Mathematics, 223, Providence, Rhode Island: American Mathematical Society, pp. 269–292, doi:10.1090/conm/223/03144