On the strong Sarkisov program

We showed that the strong Sarkisov Program of dimension $d$ can be derived from termination of specific log flips in dimension $\le d-1$. As a corollary, we show that the strong Sarkisov Program holds in dimension 4. Additionally, we prove the weak Sarkisov program with decreasing (augmented) Sarkisov degree. Finally, using the theory of syzygies of Mori fibre spaces, we obtain a directed diagram that encodes all the information of the strong Sarkisov Program.