Affine and Toric Hyperplane Arrangements

Abstract

We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.

Manifold arrangements

“…In the manifold setting the computation of the cd-index now depends upon the intersection poset and the Euler characteristic of the elements of this quasi-graded poset. This extends the earlier studied spherical and toric arrangements [4,15].…”

“…This is the approach taken in the paper [15] when studying toric arrangements. However, this does not work for spherical arrangements since the zero-dimensional sphere consists of two points and hence is disconnected.…”

“…Their work shows how to determine the cd-index of induced subdivisions of the sphere S n . Ehrenborg, Readdy and Slone considered toric arrangements that induce regular subdivisions of the torus T n [15]. Yet again, the associated cd-index depends only upon the flag f -vector of the intersection poset.…”

“…The previous proof techniques for studying subdivisions induced by oriented matroids and toric arrangements depended upon finding a natural map from the face poset of the given subdivision to the intersection poset and studying the inverse image of a chain under this map. See the proofs of [3,Theorem 4.5], [4,Theorem 3.1] and [15,Theorems 3.12 and 4.10]. In the more general setting of manifold arrangements we can now avoid this step by forming another Whitney stratification having the same cd-index; see Proposition 7.6.…”

“…In both of these cases Theorem 7.3 reduces to a result which only depends on the intersection poset. The original work for spherical and toric arrangements required the induced subdivision to yield a regular subdivision on the sphere S n , respectively, the torus T n [4,15]. This regularity condition is no longer necessary in the arena of Whitney stratified spaces.…”