We consider ℙ(1, 1, 1, 2) bundles over ℙ1 and construct hypersurfaces of these bundles which form a degree 2 del Pezzo fibration over ℙ1 as a Mori fibre space. We classify all such hypersurfaces whose type III or IV Sarkisov link, inherited from the ambient space, pass to a different Mori fibre space. A similar result for cubic surface fibrations over ℙ2 is also presented.

We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kähler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) compute the stability thresholds for hypersurfaces at generalised Eckardt points and for cubic surfaces at all points, and (c) provide a new algebraic proof of Tian’s criterion for K-stability, amongst other applications.

Abstract. We highlight a relation between the existence of Sarkisov links and the finite generation of (certain) Cox rings. We introduce explicit methods to use this relation in order to prove birational rigidity statements. To illustrate, we complete the birational rigidity results of Okada for Fano complete intersection 3-folds in singular weighted projective spaces.

Abstract. 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 DeligneMumford 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 P 1 , into other Mori fibre spaces, using Cox rings and variation of geometric invariant theory. We show that the pliability of these Mori fibre spaces is at least three and they are not rational.

We classify birationally rigid orbifold Fano 3-folds of index 1 defined by 5 × 5 Pfaffian varieties. We give a sharp criterion for the birational rigidity of these families based on the type of singularities that the varieties admit. Various conjectures are born out of our study, highlighting a possible approach to the classification of terminal Fano 3-folds. The birationally rigid cases are the first known rigid examples of Fano varieties that are not (weighted) complete intersections.