Algebraic geometry seminar in Higher School of Economics

February 2016, Moscow, Russia

Title: Rationally conneted non Fano type vareties

Abstract: The class of varieties of Fano type is a singular generalization of Fano varieties which is very well behaved under the MMP, the Cox ring of varietie of Fano type is finitely generated. It is known that all varieties of Fano type are rationally connected. The convers is true in a sense in dimension 2. I will give counterexamples to the converse statement in dimension 3 and higher. I will use the techniques of birational rigidity and the MMP.

5th Swiss-French workshop on algebraic geometry

January 2016, Charmey, Switzerland

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

Abstract: Study of embeddings of a finite group G into the Cremona group is equivalent to study of G-birational geometry of rational G-Mori fiber spaces. A good place to start study finite subgroups is a study of simple subgroup. We 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 Xn 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 Xn are superrigid, in particular not rational, for n>2.

University of Liverpool Geometry seminar

December, 2015, Liverpool, 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.

Edinburgh Geometry seminar

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.

Conference on Finite subgroups of Cremona group

August, 2015, Trento, Italy

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.

Algebraic geometry seminar in IBS-CGP

June, 2015, Pohang, Korea

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.

EDGE days

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.