Geometry of arithmetic surfaces

Abstract

In this thesis my emphasis is on the resolution of the singularities of fibre products of Arithmetic Surfaces. In chapter one as an introduction to my thesis some elementary concepts related to regular and singular points are reviewed and the concept of tangent cone is defined for schemes over a discrete valuation ring. The concept of arithmetic surfaces is introduced briefly in the end of this chapter. In chapter 2 my new procedures namely the procedure of Mojgan(_1) and the procedure of Mahtab(_2) and a new operator called Moje are introduced. Also the concept of tangent space is defined for schemes over a discrete valuation ring. In chapter 3 the singularities of schemes which are the fibre products of some surfaces with ordinary double points are resolved. It is done in two different methods. The results from both methods are consistent. In chapter 4, I have tried to resolve the singularities of a special class of arithmetic three-folds, namely those which are the fibre product of two arithmetic surfaces, which were very helpful to achieve my final results about the resolution of singularities of fibre products of the minimal regular models of Tate. Chapter 5 includes my final results which are about the resolution of singularities of the fibre product of two minimal regular models of Tate.