Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 080, 19 pages      arXiv:1908.01535      https://doi.org/10.3842/SIGMA.2020.080
Contribution to the Special Issue on Primitive Forms and Related Topics in honor of Kyoji Saito for his 77th birthday

Modular Construction of Free Hyperplane Arrangements

Shuhei Tsujie
Department of Education, Hokkaido University of Education, Hokkaido, Japan

Received January 29, 2020, in final form August 13, 2020; Published online August 22, 2020

Abstract
In this article, we study freeness of hyperplane arrangements. One of the most investigated arrangement is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal and Dirac showed that a graph is chordal if and only if the graph is obtained by ''gluing'' complete graphs. We will generalize Dirac's construction to simple matroids with modular joins introduced by Ziegler and show that every arrangement whose associated matroid is constructed in the manner mentioned above is divisionally free. Moreover, we apply the result to arrangements associated with gain graphs and arrangements over finite fields.

Key words: hyperplane arrangement; free arrangement; matroid; modular join; chordality.

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