March, 2016, Edinburgh, UK

**Title:** Classification and birational rigidity of del Pezzo fibrations with an action of the Klein simple group

**Abstract:** Cremona group of rank b is the group of birational transformations of the projective n-space. One was to study Cremona group is to study its finite subgroups. This problem can be translated to the geometric language: instead of subgroups of Cremona group isomorphic to a group G we can study rational G-Mori fiber spaces. This idea works particularly well for simple subgroups of Cremona group. I prove that any del Pezzo fibration over projective line with an action of the Klein simple group is either a direct product or a certain singular del Pezzo fibration X_{n} of degree 2. It is known that del Pezzo fibrations of degree 2 satisfying the K^{2}-condition are birationally superrigid. I extend this result to singular del Pezzo fibrations and prove that X_{n} are superrigid, in particular not rational, for n>2.