Topology of Moduli Spaces and Complements of Hyperplane Arrangements

A complex l-arrangement A is a finite collection of hyperplanes in a l-dimensional affine (or projective) space. The study of the interplay between the topology of the complement and combinatorial structure of an arrangement is one of the central topics in combinatorics, topology and algebraic geometry. My thesis aims to address the following natural questions: 1. When the combinatorial structure of a hyperplane arrangement will determine the topology of its complement? 2. To what extend the combinatorial structure of a fiber-type projective line arrangement will determine the topology of its complement? 3. When is the moduli space of line arrangements with fixed combinatorial structure connected? I partially answer the first question by showing that if the arrangement A is nice or nice point then the moduli space is connected with respect to the fixed combinatorial structure of A. Fiber-type projective line arrangements are also studied towards the second question. I also find a counter example of two fiber-type projective line arrangements with same combinatorial structure but disconnected moduli space. However the fundamental groups of complements of the arrangements in the example are isomorphic. Toward to the third question, I classified arrangements of nine lines.