The Characteristic Polynomials Of Plane Arrangements

In this thesis, using the algebraic-geometric method, we study characteristic polynomial of plane arrangements in three dimensional vector space.Firstly, We compute the characteristic polynomials of plane arrangements by the deconing construction of the arrangements. We obtain the general formula of the characteristic polynomials of a class plane arrangement, and the general term formulas of simple pyramid and prism.Secondly, we the classify plane arrangements with at most 5 planes by the characteristic polynomial. We get 5 classes of central plane arrangements and17 classes of non-central plane arrangements.Thirdly, using the Deletion-Contraction theorem and generalized the Deletion-Contraction theorem to the signed graphic arrangements, we calculate the chromatic polynomials of graphic arrangements including simply graphic arrangements and signed graphic arrangements. Some general formulas of graphic arrangements are found.