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
editFor 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
editFor 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 defined by rational maps.
Let 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
editTheorem (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
editMnëv'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]
See also
edit- 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.
References
edit- ^ Mnëv, N. E. (1988), "The universality theorems on the classification problem of configuration varieties and convex polytopes varieties", in Viro, Oleg Yanovich; Vershik, Anatoly Moiseevich (eds.), Topology and geometry—Rohlin Seminar, Lecture Notes in Math., vol. 1346, Springer, Berlin, pp. 527–543, doi:10.1007/BFb0082792, MR 0970093
- ^ Vershik, A. M. (1988), "Topology of the convex polytopes' manifolds, the manifold of the projective configurations of a given combinatorial type and representations of lattices", in Viro, Oleg Yanovich; Vershik, Anatoly Moiseevich (eds.), Topology and geometry—Rohlin Seminar, Lecture Notes in Math., vol. 1346, Springer, Berlin, pp. 557–581, doi:10.1007/BFb0082794, MR 0970095
- ^ Kapovich, Michael; Millson, John J. (1999), Brylinski, Jean-Luc; Brylinski, Ranee; Nistor, Victor; Tsygan, Boris (eds.), "Moduli Spaces of Linkages and Arrangements", Advances in Geometry, Boston, MA: Birkhäuser, pp. 237–270, doi:10.1007/978-1-4612-1770-1_11, ISBN 978-1-4612-1770-1, retrieved 2023-04-17
Further reading
edit- 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