Birational Relation Among Mori Fiber Spaces

The purpose of this chapter is to discuss the birational relation among Mori fiber spaces. Here we focus our attention on the most important subject, the Sarkisov program, due to Sarkisov [3], Reid [6], Corti [1], which gives an algorithm for factoring a given birational map between Mori fiber spaces into a sequence of certain elementary transformations called “links.” While it is a higher-dimensional analogue of the Castelnuovo—Noether theorem (cf. Theorem 1-8-8), its true meaning becomes clearer in the framework of the logarithmic category, with the main machinery of the program working under the log MMP discussed in Chapter 11. Our presentation is mostly in dimension 3, where all the necessary ingredients are established (with the most subtle part of showing “termination of Sarkisov program” ingeniously settled by Corti [1], as discussed in Section 13–2), leaving the details of the higher-dimensional case to the reader, where the general mechanism goes almost verbatim but some key ingredients still remain conjectural. (See Section 14–5 for the toric Sarkisov program, where we have all the necessary ingredients established in all dimensions.)