Arrangement of hyperplanes


In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set A of hyperplanes in a linear, affine, or projective space S.
Questions about a hyperplane arrangement A generally concern geometrical, topological, or other properties of the complement, M, which is the set that remains when the hyperplanes are removed from the whole space. One may ask how these properties are related to the arrangement and its intersection semilattice.
The intersection semilattice of A, written L, is the set of all subspaces that are obtained by intersecting some of the hyperplanes; among these subspaces are S itself, all the individual hyperplanes, all intersections of pairs of hyperplanes, etc.. These intersection subspaces of A are also called the flats of A. The intersection semilattice L is partially ordered by reverse inclusion.
If the whole space S is 2-dimensional, the hyperplanes are lines; such an arrangement is often called an arrangement of lines. Historically, real arrangements of lines were the first arrangements investigated. If S is 3-dimensional one has an arrangement of planes.

General theory

The intersection semilattice and the matroid

The intersection semilattice L is a meet semilattice and more specifically is a geometric semilattice. If the arrangement is linear or projective, or if the intersection of all hyperplanes is nonempty, the intersection lattice is a geometric lattice.
When L is a lattice, the matroid of A, written M, has A for its ground set and has rank function r := codim, where S is any subset of A and I is the intersection of the hyperplanes in S. In general, when L is a semilattice, there is an analogous matroid-like structure called a semimatroid, which is a generalization of a matroid, but is not a matroid if L is not a lattice.

Polynomials

For a subset B of A, let us define f := the intersection of the hyperplanes in B; this is S if B is empty.
The characteristic polynomial of A, written pA, can be defined by
summed over all subsets B of A except, in the affine case, subsets whose intersection is empty. This polynomial helps to solve some basic questions; see below.
Another polynomial associated with A is the Whitney-number polynomial wA, defined by
summed over B ⊆ C ⊆ A such that f is nonempty.
Being a geometric lattice or semilattice, L has a characteristic polynomial, pL, which has an extensive theory. Thus it is good to know that pA = yi pL, where i is the smallest dimension of any flat, except that in the projective case it equals yi + 1pL.
The Whitney-number polynomial of A is similarly related to that of L.

The Orlik–Solomon algebra

The intersection semilattice determines another combinatorial invariant of the arrangement, the Orlik–Solomon algebra. To define it, fix a commutative subring K of the base field and form the exterior algebra E of the vector space
generated by the hyperplanes.
A chain complex structure is defined on E with the usual boundary operator.
The Orlik–Solomon algebra is then the quotient of E by the ideal generated by elements of the form for which have empty intersection, and by boundaries of elements of the same form for which has codimension less than p.

Real arrangements

In real affine space, the complement is disconnected: it is made up of separate pieces called cells or regions or chambers, each of which is either a bounded region that is a convex polytope, or an unbounded region that is a convex polyhedral region which goes off to infinity.
Each flat of A is also divided into pieces by the hyperplanes that do not contain the flat; these pieces are called the faces of A.
The regions are faces because the whole space is a flat.
The faces of codimension 1 may be called the facets of A.
The face semilattice of an arrangement is the set of all faces, ordered by inclusion. Adding an extra top element to the face semilattice gives the face lattice.
In two dimensions each region is a convex polygon or a convex polygonal region which goes off to infinity.
Typical problems about an arrangement in n-dimensional real space are to say how many regions there are, or how many faces of dimension 4, or how many bounded regions. These questions can be answered just from the intersection semilattice. For instance, two basic theorems, from Zaslavsky, are that the number of regions of an affine arrangement equals npA and the number of bounded regions equals npA. Similarly, the number of k-dimensional faces or bounded faces can be read off as the coefficient of xn−k in n wA or nwA.
designed a fast algorithm to determine the face of an arrangement of hyperplanes containing an input point.
Another question about an arrangement in real space is to decide how many regions are simplices. This cannot be answered based solely on the intersection semilattice. The McMullen problem asks for the smallest arrangement of a given dimension in general position in real projective space for which there does not exist a cell touched by all hyperplanes.
A real linear arrangement has, besides its face semilattice, a poset of regions, a different one for each region. This poset is formed by choosing an arbitrary base region, B0, and associating with each region R the set S consisting of the hyperplanes that separate R from B. The regions are partially ordered so that R1 ≥ R2 if S contains S. In the special case when the hyperplanes arise from a root system, the resulting poset is the corresponding Weyl group with the weak Bruhat order. In general, the poset of regions is ranked by the number of separating hyperplanes and its Möbius function has been computed.
Vadim Schechtman and Alexander Varchenko introduced a matrix indexed by the regions. The matrix element for the region and is given by the product of indeterminate variables for every hyperplane H that separates these two regions. If these variables are specialized to be all value q, then this is called the q-matrix for the arrangement and much information is contained in its Smith normal form.

Complex arrangements

In complex affine space, the complement is connected with holes where the hyperplanes were removed.
A typical problem about an arrangement in complex space is to describe the holes.
The basic theorem about complex arrangements is that the cohomology of the complement M is completely determined by the intersection semilattice. To be precise, the cohomology ring of M is isomorphic to the Orlik–Solomon algebra on Z.
The isomorphism can be described explicitly and gives a presentation of the cohomology in terms of generators and relations, where generators are represented as logarithmic differential forms
with any linear form defining the generic hyperplane of the arrangement.

Technicalities

Sometimes it is convenient to allow the degenerate hyperplane, which is the whole space S, to belong to an arrangement. If A contains the degenerate hyperplane, then it has no regions because the complement is empty. However, it still has flats, an intersection semilattice, and faces. The preceding discussion assumes the degenerate hyperplane is not in the arrangement.
Sometimes one wants to allow repeated hyperplanes in the arrangement. We did not consider this possibility in the preceding discussion, but it makes no material difference.