Plus-one Generated and Next to Free Arrangements of Hyperplanes

Author:

Abe Takuro1

Affiliation:

1. Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan

Abstract

Abstract 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.

Funder

JSPS Grant-in-Aid for Scientific Research

JSPS Grant-in-Aid for Exploratory Research

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

Reference25 articles.

1. Roots of characteristic polynomials and intersection points of line arrangements;Abe;J. Singularities,2014

2. Divisionally free arrangements of hyperplanes;Abe;Invent. Math.,2016

3. Restrictions of free arrangements and the division theorem;Abe;Proceedings of the Intensive Period “Perspectives in Lie Theory”,2017

4. Deletion theorem and combinatorics of hyperplane arrangements;Abe;Math. Ann.,2019

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