On del Pezzo fibrations in positive characteristic

Fibrations play a key role in the classification problems of algebraic varieties. While in characteristic zero, the general fibre of a morphism between smooth varieties is still smooth, this is no longer true in general over fields of positive characteristic (where the classical examples are quasi-elliptic fibrations). However, one can hope to bound such a bad behaviour to small primes if the generic fibre has special global properties. In this talk, I will discuss a joint work with H. Tanaka where we study the special case of Mori fibrations in positive characteristic, which constitute one of the possible outcomes of the Minimal Model Program. Our results are a consequence of a detailed study of the possible pathologies appearing on log del Pezzo surfaces over imperfect fields.