Del Pezzo fibrations and rank 3 Cox rings

One possible output of the minimal model program is a Mori fibre space. These varieties in 3 dimensions are Fano varieties, del Pezzo fibrations and conic bundles. Uniqueness of this output, the so-called rigidity of a Mori fibre space, is a question which arises naturally. In many cases, it has been proven for a general Fano 3-fold to be rigid. Del Pezzo fibrations over the rational curve have been studied in higher degrees and consequently it is known that if deg > 3 then the del Pezzo fibration is nonrigid.

The goal of this thesis is to study rigidity and nonrigidity of low degree del Pezzo fibrations. We give a construction of these objects and classify the nonrigid ones whose link to the other model is obtained by the ambient space. This, in particular, provides many examples of nonrigid degree 2 del Pezzo fibrations which are not necessarily smooth. It is known that the study of rigidity for degree 3 del Pezzo fibrations is subject to consideration of Corti-KoMr stability condition. A first attempt to generalise this stability notion for lower degree fibrations is given in this thesis. The relation between these stability conditions and Sarkisov program is also studied in an explicit way. This requires techniques of working with rank 3 Cox rings which we develop. In particular, the notion of well-formedness for Cox rings is introduced as a generalisation of well- formedness of weighted projective spaces. We also construct families of cubic surface fibration in dimension 4 and study their nonrigidity in a similar way.