On the number and boundedness of minimal models of a variety of general type

Finding minimal models is the first step in the birational classification of smooth projective varieties. After it is established that a minimal model exists some natural questions arise such as: is it the minimal model unique? If not, how many are they? After recalling all the necessary notions of the Minimal Model Program, I will explain that varieties of general type admit a finite number of minimal models. I will talk about a recent joint project with Stefan Schreieder and Luca Tasin where we prove that this number is bounded by a constant depending only on the canonical volume. It follows that in any dimension, minimal models of general type and bounded volume form a bounded family. I will also show that in some cases for threefolds, it is possible to compute this constant explicitly.