June, 2016, Edinburgh, UK

Title: Rationality and Stable rationality of Del Pezzo fibrations

Abstract: I consider stable rationality of double covers of P^m-bundles over projective spaces branched over divisors of high enough degree. In particular I study del Pezzo fibrations of degree 2 with a smooth total space. I prove that a very general del Pezzo fibration of degree 2 is not stably rational. Then I discuss the case of singular total space, which singularities does it make sense to consider, and discuss my non rationality result for this class of varieties.

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 K²-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.

October, 2015, Edinburgh, UK

Title: Rationally connected non Fano type varieties

Abstract: The class of varieties of Fano type is a generalization of Fano varieties which is very well behaved under the MMP. It is known that all varieties of Fano type are rationally connected. The converse is true in a sense in dimension 2. I will give counterexamples in dimension 3 and higher using the technique of singularities of linear systems which is typically used for proving birational rigidity.

June, 2015, Edinburgh, UK

Title: Rationally connected non Fano type varieties

Abstract: The class of varieties of Fano type is a generalization of Fano varieties which is very well behaved under the MMP. It is known that all varieties of Fano type are rationally connected. The converse is true in a sense in dimension 2. I will give counterexamples in dimension 3 and higher using the technique of singularities of linear systems which is typically used for proving birational rigidity.