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? How do they behave in families? 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. 

Then I will talk about a recent joint project with Stefan Schreieder and Luca Tasin where we prove that minimal models of general type of given dimension and bounded volume form a bounded family.

Moreover, we prove that the number of minimal models can be bounded by a constant depending only on the canonical volume. In the end, I will show that in some cases for threefolds it is possible to give some effective bounds for this number.