On the birational geometry of Fano threefold complete intersections

The Minimal Model Program (MMP) is a far reaching conjecture in birational geometry which aims at constructing a good representative (minimal model) of any given complex projective variety W. When such a model exists it might not be unique and so it becomes natural to study the relations between them. In the case when W is covered by rational curves, its minimal model is a Mori fibre space, that is, a fibration whose generic fibre is positively curved, and its uniqueness is encoded in the notion of birational rigidity. In this talk we will give an introduction to the ideas of the MMP with the background of Fano threefold complete intersections.