Februray 2017, Moscow, Russia

Title: Stable rationality of del Pezzo fibrations.

Abstract: We say that f:X \to Z is a del Pezzo fibration if a generic fiber of f is a del Pezzo surface.

I will:

• Discuss stable rationality of del Pezzo fibrations of low degree.
• Show that a very general del Pezzo fibration of degrees 1,2, or 3, such that its anticanonical class is not ample, is not stably rational.
• Make a small survey of known results on stable rationality and about how can one improve the result on del Pezzo fibrations in dimension three.
• Discuss the mthod of reduction to finite characteristic and how to apply it for del Pezzo fibrations.

Februray 2017, Moscow, Russia

Title: Rationality of del Pezzo fibrations and the Cremona group

Abstract: I will talk about embeddings of PSL₂(7) into the Cremona group of rank 3. Study of embeddings of a finite group G into the Cremona group is equivalent to study of G-equivariant birational geometry of rational GQ-Mori fiber spaces.
Thus to classify embeddings of PSL₂(7) one has to classify rational PSL₂(7) Q-Mori fiber spaces up to PSL₂(7)-equivariant birational equivalence.
I will discuss classification of PSL₂(7) Q-del Pezzo fibrations and their birational rigidity, in particular, rationality.

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.