## Ken-Ichiro Arima : COE Lecture Series

### An introduction to the minimal model program and its extension to the case in positive characteristic

To studying the structure of algebraic varieties, running minimal model program (MMP) brings us an effective clue. The beginning of this program consists of the following two steps: Find a good representative in a fixed birational equivalence class. Study the properties of the good representative. We are able to have the properties of the first algebraic variety.

I give an overview of two or three-dimensional MMP. Two-dimensional case is classically known but three-dimensional case has many difficult problems. I introduce some of them, comparing two-dimensional case. Classification of singularity is one of these obstacles. We also study the MMP from this point of view.

Extensions of the MMP is the second topic. Here I briefly explain the log MMP and the case in positive characteristic, which is my research sphere.