Chambers of 2-affine arrangements and freeness of 3-arrangements

Freeness and Multirestriction of Hyperplane Arrangements

Generalizing a result of Yoshinaga in dimension 3, we show that a central hyperplane arrangement in 4-space is free exactly if its restriction with multiplicities to a fixed hyperplane of the arrangement is free and its reduced characteristic polynomial equals the characteristic polynomial of this restriction. We show that the same statement holds true in any dimension when imposing certain tam...

Affine and toric arrangements

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 toric arrangements, we also generalize Zaslavsky’s fundamental results on the number of regions. Résumé. Nous étendons l’opérateur de Billera–Ehrenborg–Readdy entre la trellis d’intersection et la trellis de faces d’un ...

Supersolvability and Freeness for ψ-Graphical Arrangements

Let G be a simple graph on the vertex set {v1, . . . , vn} with edge set E. Let K be a field. The graphical arrangement AG in K n is the arrangement xi − xj = 0, vivj ∈ E. An arrangement A is supersolvable if the intersection lattice L(c(A)) of the cone c(A) contains a maximal chain of modular elements. The second author has shown that a graphical arrangement AG is supersolvable if and only if ...

Affine and Toric Hyperplane Arrangements

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.

Logarithmic Forms on Affine Arrangements