Plus-one Generated and Next to Free Arrangements of Hyperplanes

By Takuro Abe. International mathematics research notices, 2018

66 citations recorded by the source corpus.

We introduce a new class of arrangements of hyperplanes, called (strictly) plus-one generated arrangements, from algebraic point of view. Plus-one generatedness is close to freeness, that is, plus-one generated arrangements have their logarithmic derivation modules generated by dimension plus-one elements, with relations containing one linear form coefficient. We show that strictly plus-one generated arrangements can be obtained if we delete a hyperplane from free arrangements. We show a relative freeness criterion in terms of plus-one generatedness. In particular, for plane arrangements, we show that a free arrangement is in fact surrounded by free or strictly plus-one generated arrangements. We also give several applications.

Subfields: Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f0748ac8c969a91df3d99559d731bce0134dd2ad