MINIMAL MODEL PROGRAM AND THE JORDAN PROPERTY OF AUTOMORPHISM GROUPS OF VARIETIES

Abstract

The aim of this thesis is to give an introduction and expository account into two active areas of research in the field of algebraic geometry. The first area will on the well-known Minimal Model Program (MMP) (also known as Mori’s Program). We aim to give a bird-eye view of the MMP, dis- cussing its motivations, problems arising from constructing the MMP as well as several important theorems resulting from the study of the MMP. The four big theorems of the MMP: The Non-Vanishing theorem, The Rationality theorem, The Basepoint-free theorem as well as the Cone theorems will be discussed in greater detail. This will consist the bulk of the thesis. The second area will be on the Jordan property of groups. In particular, we re- strict our scenario to the case where our group is of the form Aut(X) or Bir(X) for certain algebraic varieties X. Due to time constraints, it is not possible to go into the proofs and details of every known result. Instead, we will attempt to give a quick overview on the recent developments and highlight cases that are still currently left unsolved.