Relations in the Sarkisov program (Q2860718)

Let \(Z\) be a smooth complex projective variety such that \(K_Z\) is not pseudo-effective, then by \textit{C. Birkar} et al. [J. Am. Math. Soc. 23, No. 2, 405--468 (2010; Zbl 1210.14019)], it is known that there is a sequence of elementary birational maps (flips and divisorial contractions) NEWLINE\[NEWLINEZ=X_0\dasharrow X_1 \dasharrow \ldots \dasharrow X_n=XNEWLINE\]NEWLINE such that \(X\) admits the structure of a Mori fiber space \(p:X\to S\) where \(X\) is terminal and \(\mathbb Q\)-factorial, \(K_X\) is ample over \(S\) and \(\rho (X/S)=1\). The above sequence of birational maps is not uniquely determined and so a given \(Z\) may give rise to two (or more) Mori fiber spaces \(X\to S\) and \(Y\to T\) in which case we say that they are Sarkisov related. If two Mori fiber spaces \(X_i/S_i\) and \(X_j/S_j\) are obtained by running a \(K_Z/W\) minimal model program with \(\rho (X/W)=2\) then we say that \(L_{i,j}:X_i/S_i\dasharrow X_j/S_j\) is an elementary Sarkisov link. It is known that any two Sarkisov related Mori fiber spaces can be connected by a finite sequence of elementary Sarkisov links [\textit{C. D. Hacon} and \textit{J. McKernan}, J. Algebr. Geom. 22, No. 2, 389--405 (2013; Zbl 1267.14024)]. Two such sequences of elementary Sarkisov links define a relation in the Sarkisov Program. In this paper the author defines elementary relations (in the Sarkisov Program) which are obtained by running a \(K_Z/W\) minimal model program with \(\rho (Z/W)=3\) and shows that the relations in the Sarkisov Program are generated by the elementary relations.