## Birational geometry of Fano fibrations

##### Abstract

An algebraic variety is called rationally connected if two generic points can be connected by a curve isomorphic to the projective line. The output of the minimal model program applied to rationally connected variety is variety admitting Mori fiber spaces over a rationally connected base. In this thesis we study the birational geometry of a particular class of rationally connected Mori fiber spaces: Fano fibrations over the projective line. We construct examples of Fano fibrations with a unique Mori fiber space in their birational classes. We prove that these examples are not birational to varieties of Fano type, thus answering the question of Cascini and Gongyo. That is we prove that the classes of rationally connected varieties and varieties of Fano type are not birationally equivalent. To construct the examples we use the techniques of birational rigidity. A Mori fiber space is called birationally rigid if there is a unique Mori fiber space structure in its birational class. The birational rigidity of smooth varieties admitting a del Pezzo fibration of degrees 1 and 2 is a well studied question. Unfortunately it is not enough to study smooth del Pezzo fibrations as there are fibrations which do not have smooth or even smoothable minimal models. In the case of fibrations of degree 2 we know that there is a minimal model with 2-Gorenstein singularities. These singularities are degenerations of the simplest terminal quotient singularity: singular points of the type 1/2(1,1,1). We give first examples of birationally rigid del Pezzo fibrations with 2-Gorenstein singularities. We then apply this result to study finite subgroups of the Cremona group of rank three. We then study the birational geometry of Fano fibrations from a different side. Using the reduction to characteristic 2 method we prove that double covers of Pn-bundles over Pm branched over a divisor of sufficiently high degree are not stably rational. For a del Pezzo fibration Y→P1 of degree 2 such that X is smooth there is a double cover Y→ X, where X is a P2-bundle over P1. In this case a stronger result holds: a very general Y with Pic(Y)≅Z⊕Z is not stably rational. We discuss the proof of this statement.