6 - The Sarkisov Category

Summary

We shall proceed to the study of threefold Mori fibre spaces. This chapter is an introduction to the Sarkisov program, which factorises a birational map of Mori fibre spaces as a composite of elementary maps called Sarkisov links. Corti established it in dimension three. The program twists the birational map into an isomorphism by means of the Noether-Fano inequality. Whereas Hacon and McKernan obtained a new program in an arbitrary dimension, we stick to the traditional approach which fits better into the explicit study. The program stems from the pioneer work of Iskovskikh and Manin on the irrationality of a smooth quartic threefold. We review the rationality problem and discuss the notions of rationality, stable rationality and unirationality. A rational variety has a large number of Mori fibre spaces in its birational class. Oppositely, a variety birational to essentially only one Mori fibre space is said to be birationally rigid. We explain a general strategy for applying the Sarkisov program to the rationality and birational rigidity problem, and demonstrate this by an example due to Corti and Mella which has exactly two birational structures of a Mori fibre space.