Liverpool algebraic geometry seminar

Februray 2017, Moscow, Russia

Title: Birational geometry of del Pezzo fibrations

Abstract: Del Pezzo brations appear as minimal models of rationally connected varieties. The rationality of smooth del Pezzo brations is a well studied question but smooth brations are not dense in moduli. Little is known about the rationality of the singular models. We prove
birational rigidity, hence non-rationality, of del Pezzo brations with simple non-Gorenstein singularities satisfying the famous K2-condition. We then apply this result to study embeddings of PSL2(7) into the Cremona group.

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.