Modular Elements In A Class Of Graphical Arrangements

The main topic of this thesis is about central hyperplane arrangements with modular elements in their intersection lattice. Chapter 1 is a brief introduction of the work. The arrangement of hyperplanes and modular elements are mainly introduced in chapter 2. In chapter 3, a theorem for graphic arrangements with modular elements is given. This theorem is one of the main results of this thesis. It proved that the intersection of hyperplanes corresponding to the edges of a clique in a simple graph is a modular. As a corollary of Stanley's theorem, a factorization of the Poincare polynomial over integers of the graphic arrangement is obtained and described in detail. If X is a modular element of A, the M(o|¨)bius function in the factorization of the Poincare polynomial can be obtained from the partial Hasse diagram. There is given an example in which X is not modular even if X is the intersection of hyperplanes corresponding to the edges of a chordal subgraph. In chapter 4, the...