On pliability of del Pezzo fibrations and Cox rings

We develop some concrete methods to build Sarkisov links, starting from Mori ?fibre spaces. This is done by studying low rank Cox rings and their properties. As part of this development, we give an algorithm to construct explicitly the coarse moduli space of a toric Deligne-Mumford stack. This can be viewed as the generalisation of the notion of well-formedness for weighted projective spaces to homogeneous coordinate ring of toric varieties. As an illustration, we apply these methods to study birational transformations of certain ?fibrations of del Pezzo surfaces over $\mathbb{P}^1$, into other Mori ?fibre spaces, using Cox rings and variation of geometric invariant theory. We show that the pliability of these Mori fi?bre spaces is at least three and they are not rational.