Mnëv's universality theorem explained
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
is a list of all partitions of points in
induced by hyperplanes in
. In particular, the structure of oriented matroid contains full information on the incidence relations in
, inducing on
a
matroid structure.
The realization space of an oriented matroid is the space of all configurations of points
inducing the same oriented matroid structure on
.
Stable equivalence of semialgebraic sets
For the purposes of universality, the stable equivalence of semialgebraic sets is defined as follows.
Let
and
be semialgebraic sets, obtained as a disconnected union of connected semialgebraic sets
We say that
and
are
rationally equivalent if there exist homeomorphisms
Ui\stackrel{\varphii}\mapstoVi
defined by rational maps.
Let
U\subset{R}n+d,V\subset{R}n
be semialgebraic sets,with
mapping to
under the natural projection
deleting the last
coordinates. We say that
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.
Mnëv's universality theorem
Theorem (Mnëv's universality theorem):
Let
be a semialgebraic subset in
defined over integers. Then
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 has been used also by Kapovich and Millson in the study of the moduli spaces of linkages and arrangements.
See also
- 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.