hyperplane arrangement

Let $V$ be a vector space over a field $\mathbb{K}$ . A hyperplane arrangment in $V$ is a family $\mathcal{A}=\{\mathcal{H}_{i}\}_{i\in I}$ of affine hyperplanes in $V$ . If all of the hyperplanes pass through $0$ , $\mathcal{A}$ 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 $V=\mathbb{K}^{n}$ . Then the family

\mathbb{K}P^{n}=\{S\subset V\mid\dim_{\mathbb{K}}(S)=1\}

of $1$ -dimensional subspaces of $V$ is a central subspace arrangement, the projective space of dimension $n$ over $\mathbb{K}$ .

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

Example 2.

Again let $V=\mathbb{K}^{n}$ , and suppose $0\leq k\leq n$ . Then the family

\operatorname{Gr}(V,k)=\{S\subset V\mid\dim_{\mathbb{K}}(S)=k\}

of $k$ -dimensional subspaces of $V$ is a central subspace arrangement, the Grassmannian. Observe that $\mathbb{K}P^{n}=\operatorname{Gr}(\mathbb{K}^{n},1)$ .

If $V$ is a topological vector space and $\mathcal{A}$ is a hyperplane arrangement, then it makes sense to ask for the fundamental group of the complement $V\setminus\bigcup_{\mathcal{H}\in\mathcal{A}}\mathcal{H}$ .

Example 3.

If $\mathcal{A}$ is a finite hyperplane arrangement over $V=\mathcal{R}^{n}$ , 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 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 $\mathbb{R}^{n}\hookrightarrow\mathbb{C}^{n}$ to complexify $\mathcal{A}$ . In this case the complement $\mathbb{C}^{n}\setminus\bigcup_{\mathcal{H}\in\mathcal{A}}\mathcal{H}$ usually has nontrivial fundamental group.

References

1 Klain, D. A., and G.-C. Rota, , Introduction to geometric probability, Cambridge University Press, 1997.
2 Orlik, P., and H. 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