hyperplane arrangement
Let be a vector space over a field . A hyperplane arrangment in is a family of affine hyperplanes in . If all of the hyperplanes pass through , is called central; otherwise, it is affine. More generally, a subspace arrangement is a family of affine subspaces of . The same distinction between central and affine subspace arrangement holds.
Example 1.
Let . Then the family
of -dimensional subspaces of is a central subspace
arrangement, the projective space of dimension over
.
Instead of considering all lines through a vector space, we could consider all -dimensional subspaces of the space.
Example 2.
Again let , and suppose . Then the family
of -dimensional subspaces of is a central subspace arrangement, the Grassmannian. Observe that .
If is a topological vector space and is a
hyperplane arrangement, then it makes sense to ask for the
fundamental group
of the complement
.
Example 3.
If is a finite hyperplane arrangement over
, then the arrangement
partitions (http://planetmath.org/Partition)
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 equivalent to the
zonotope dual to the arrangement. Since the question of
the fundamental group here is not interesting, we could also
use the embedding
to complexify . In this case the
complement
usually has nontrivial fundamental group.
References
- 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 |