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.