A Brief Sketch of the Minimal Model Program

The pursuit of general structure theorems on algebraic varieties often leads naturally to questions of classification. One approach to such theorems is via the notion of birational equivalence, and more specifically, can be broken into the following two steps: (i) Understand the relationship between any two birationally equivalent varieties. (ii) Find natural “simplest” representatives of every birational equivalence class, called a minimal model. One can easily imagine that if such a program were carried out, it would constitute a powerful tool in proving structure theorems. Indeed, one could first prove a theorem for the minimal models provided by (ii), which are intended to be easier to work with, and then use (i) to deduce the statement for arbitrary varieties. A hint of the complexity of (ii) is that the following is already a very deep and powerful theorem: