Arrangement of hyperplanes

(This is why the semilattice must be ordered by reverse inclusion—rather than by inclusion, which might seem more natural but would not yield a geometric (semi)lattice.)

When L(A) is a lattice, the matroid of A, written M(A), has A for its ground set and has rank function r(B) := codim(f(B)), where B is any subset of A and f(B) is the intersection of the hyperplanes in B.

Being a geometric lattice or semilattice, L(A) has a characteristic polynomial, pL(A)(y), which has an extensive theory (see matroid).

(The empty set is excluded from the semilattice in the affine case specifically so that these relationships will be valid.)

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

For instance, two basic theorems, from Zaslavsky (1975), are that the number of regions of an affine arrangement equals (−1)npA(−1) and the number of bounded regions equals (−1)npA(1).

Meiser (1993) 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 (the n-dimensional generalization of triangles and tetrahedra).

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.

In the special case when the hyperplanes arise from a root system, the resulting poset is the corresponding Weyl group with the weak order.

In general, the poset of regions is ranked by the number of separating hyperplanes and its Möbius function has been computed (Edelman 1984).

Vadim Schechtman and Alexander Varchenko introduced a matrix indexed by the regions.

In complex affine space (which is hard to visualize because even the complex affine plane has four real dimensions), the complement is connected (all one piece) 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(A) is completely determined by the intersection semilattice.

To be precise, the cohomology ring of M(A) (with integer coefficients) 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 (in the de Rham cohomology) as logarithmic differential forms with