hyperplane arrangement

Let V be a vector space over a field 𝕂. A hyperplane arrangment in V is a family π’œ={β„‹i}i∈I of affine hyperplanes in V. If all of the hyperplanes pass through 0, π’œ is called central; otherwise, it is affine. More generally, a subspace arrangement is a family of affine subspaces of V. The same distinction between central and affine subspace arrangement holds.

Example 1.

Let V=Kn. Then the family


of 1-dimensional subspaces of V is a central subspace arrangement, the projective space of dimensionPlanetmathPlanetmath n over K.

Instead of considering all lines through a vector space, we could consider all k-dimensional subspaces of the space.

Example 2.

Again let V=Kn, and suppose 0≀k≀n. Then the family


of k-dimensional subspaces of V is a central subspace arrangement, the Grassmannian. Observe that K⁒Pn=Gr⁑(Kn,1).

If V is a topological vector spaceMathworldPlanetmath and π’œ is a hyperplane arrangement, then it makes sense to ask for the fundamental groupMathworldPlanetmathPlanetmath of the complement Vβˆ–β‹ƒβ„‹βˆˆπ’œβ„‹.

Example 3.

If A is a finite hyperplane arrangement over V=Rn, then the arrangement partitions (http://planetmath.org/Partition) V into a finite number of contractible cells. By selecting a point in each cell and taking the convex hull of the result, we obtain a polytope combinatorially equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the zonotope dual to the arrangement. Since the question of the fundamental group here is not interesting, we could also use the embeddingPlanetmathPlanetmathPlanetmath Rnβ†ͺCn to complexify A. In this case the complement Cnβˆ–β‹ƒH∈AH usually has nontrivial fundamental group.


  • 1 Klain, D.Β A., and G.-C.Β Rota, , Introduction to geometric probability, Cambridge University Press, 1997.
  • 2 Orlik, P., and H.Β Terao, Arrangements of hyperplanes, Springer-Verlag, 1992.
Title hyperplane arrangement
Canonical name HyperplaneArrangement
Date of creation 2013-03-22 15:47:55
Last modified on 2013-03-22 15:47:55
Owner mps (409)
Last modified by mps (409)
Numerical id 6
Author mps (409)
Entry type Definition
Classification msc 52B99
Classification msc 52C35
Synonym subspace arrangement
Synonym central arrangement
Related topic Zonotope
Defines Grassmannian