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.