Birational Geometry has been studied from the beginning of algebraic geometry. Because two birational equivalent varieties have many common properties, it is natural to find a simplest model for the classification of algebraic varieties. The minimal model program (MMP), or Mori's program is a higher dimensional version of the theory of minimal models for algebraic surfaces.



The MMP is an algorithm to find a minimal model of a given variety by constructing birational maps and simpler varieties. When we run the MMP, the main difficulty comes from the situation when the contraction map is a small contraction. To resolve this, we need a map which is called a flip. To complete the MMP, we must show that flips exist and that they terminate after finitely many steps. There are many recent works related with this, especially by Bikar, Cascini, Hacon, and McKernan.



Mori dream spaces are introduced by Hu and Keel as varieties for which the minimal model program can be run in a very good way. These spaces have a number of nice properties, and some interesting related cone structures. There are several known Mori dream spaces including log Fano varieties and spherical varieties. I will talk about several examples of Mori dream spaces, and describe what the flips look like on those spaces.