Sarkisov program for klt pairs and F-dlt foliated pairs.
Abstract
One of the goals of birational geometry is to classify varieties up to birational equivalence. The minimal model program is to find a good representative in every fixed birational equivalence class.
All of these good representatives can be divided into two classes. One is called the minimal model, and the other is called Mori fiber space, both of which are not unique.
Modulo isomorphism, any two MMP-related Mori fiber spaces are connected by a birational map, which can be decomposed into finitely many elementary birational maps by running the Sarkisov program. There are four types of elementary birational maps, called Sarkisov links.
The first goal of this paper is to introduce three methods of the Sarkisov program.
The second goal is trying to establish the Sarkisov program for foliated pairs. By reducing $F$-dlt foliated pairs to klt pairs, there is a weak decomposition of birational maps between foliated Mori fiber spaces.