Divisionally free arrangements of hyperplanes

By Takuro Abe. 2015

53 citations recorded by the source corpus.

Let $$({\mathcal {A}},{\mathcal {A}}',{\mathcal {A}}^H)$$(A,A′,AH) be the triple of hyperplane arrangements. We show that the freeness of $${\mathcal {A}}^H$$AH and the division of $$\chi ({\mathcal {A}}\,;\,t)$$χ(A;t) by $$\chi ({\mathcal {A}}^H\,;t)$$χ(AH;t) imply the freeness of $${\mathcal {A}}$$A. This “division theorem” improves the famous addition–deletion theorem, and it has several applications, which include a definition of “divisionally free arrangements”. It is a strictly larger class of free arrangements than the classical important class of inductively free arrangements. Also, whether an arrangement is divisionally free or not is determined by the combinatorics. Moreover, we show that a lot of recursively free arrangements, to which almost all known free arrangements are belonging, are divisionally free.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6f32bb04969e5ff93f4a520aa379a0755e8d4196

Citations in the corpus

• Plus-one Generated and Next to Free Arrangements of Hyperplanes (2018, unknown): “A) = (d 1,...,dℓ−1,dℓ). 8 (2) A′ is free with exp(A′) = (d 1,...,dℓ−1,dℓ −1). (3) A′′ is free with exp(A′′) = (d 1,...,dℓ−1). Moreover, all the three hold true if A and A′ are both free. Theorem 2.5 ([2], Division theorem) Let H∈ A. Then A is free if AH is free and π(AH;t) | π(A;t). The following is”