June, 2016, Edinburgh, UK

**Title:** Rationality and Stable rationality of Del Pezzo fibrations

**Abstract:** I consider stable rationality of double covers of P^{m}-bundles over projective spaces branched over divisors of high enough degree. In particular I study del Pezzo fibrations of degree 2 with a smooth total space. I prove that a very general del Pezzo fibration of degree 2 is not stably rational. Then I discuss the case of singular total space, which singularities does it make sense to consider, and discuss my non rationality result for this class of varieties.