The Supersolvability And Freeness Of A Hyperplane Arrangement

The present thesis gives an explicit construction of a basis for the derivation moduleof the cone over Shi arrangement of the type D. We obtain the relation between a super-solvable order of hyperplanes with a quadratic order of hyperplanes, and an inductivelyfree arrangement respectively. Moreover, the algorithms for calculating the characteristicpolynomial and judging the supersolvability of a central arrangement are given.Terao’s conjecture was pointed out for30years, so far it is still an open problem inthe feld of hyperplane arrangements. Recently, the main methods to solve this conjectureare the studying of multi-Coxeter arrangements. If we can understand the freeness of themulti-Coxeter arrangements thoroughly, it may contribute greatly to the settlement of theconjecture. Since there is a close relation between the cone over Shi arrangements and themulti-Coxeter arrangements, our study about the cone over Shi arrangement of the typeD gives a great help for understanding the freeness of the multi-Coxeter arrangements.The supersolvable arrangements are very important in the feld of hyperplane ar-rangements. They have good properties in Combinatorics and Topology. We study ona supersolvable order of hyperplanes deeply, and show the relation between it with aquadratic order of hyperplanes, and an inductively free arrangement respectively.The thesis is organized as follows.In Chapter1, we frst review briefy the background of the subject of hyperplanearrangements, and introduce the outline of development of it in recent years. Moreover,we introduce the structure of the full thesis and describe the main content of this thesis.In Chapter2, we mainly introduce some defnitions and conclusions of hyperplane ar-rangements, free arrangements and multiarrangements, Coxeter arrangements and multi-Coxeter arrangements. In addition, in the last section of this chapter, we introduce thebackground of Shi arrangements and give an explicit construction of a basis for the deriva-tion module of the cone over Shi arrangement of the type D. In Chapter3, we study on supersolvable arrangements, quadratic arrangements andinductively free arrangements. In the frst section, we introduce the background and def-nitions of supersolvable arrangements and quadratic arrangements. In the second section,we provide a sufcient and necessary condition for a supersolvable order of hyperplanes.In the third section, we obtain the relation between the set of supersolvable orders ofhyperplanes and the set of quadratic orders of hyperplanes. In the fourth section, weshow the relation between a supersolvable order of hyperplanes and an inductively freearrangement.In Chapter4, we obtain the algorithms for calculating the characteristic polynomialand judging the supersolvability of a central arrangement.