Relative MMP without \(\mathbb{Q}\)-factoriality (Q2055157)

Given \(g \colon (X,\Theta) \to Y\) a projective, birational morphism with reduced exceptional divisor \(E = E_1 + \cdots + E_n\) with \(E_i\)'s \(\mathbb{Q}\)-Cartier, the mmp with scaling of a divisor \(H\) with support in \(E\) satisfying \(K_X + \Theta + r_X H\) is \(g\)-ample for some \(r_X >0\), is a (finite) sequence of birational maps \(\varphi_i\colon X^i \to Z^i\), \(\varphi_{i+1}\colon X^{i+1}\to Z^i\) inducing \(\tau_i\colon X^i\dashrightarrow X^{i+1}\) over \(Y\), such that \(X^1 = X\) and \(\Theta^{i+1} = \tau_{i_*} \Theta^i\) , \(H^{i+1} = \tau_{i_*} H^i\), the divisor \(K_{X^{i+1}} + \Theta^{i+1} + r_{i+1} H_{i+1}\) is \(g_{i+1}\)-nef but not \(g_{i+1}\)-ample for some \(r_{i+1}< r_i<r_X\), and such that \(\tau_i\) is the flip or the divisorial contraction of a \((K_{X^i} + \Theta^i)\)-negative relative extremal ray \(R_i\) over \(Y\). The final goal is to find \(X^m\) such that \(K_{X^m}+\Theta^m\) is nef or \(\varphi_m\) is a Fano contraction. In most treatments \(X\) is replaced by a birational modification so that \(K_X+\Theta\) is \(\mathbb{Q}\)-Cartier since otherwise a divisorial contraction may have to be replaced by a small modification resulting in the so-called \textsl{mixed case}. In this work the author deals with a case when \(X\) is not necessarily \(\mathbb{Q}\)-factorial. Assuming that the coefficients of \(H=\sum_ih_iE_i\) are non-zero, negative and linearly independent over \(\mathbb{Q}(e_1, \cdots, e_n)\) where \(K_X+\Theta \equiv_g \sum_ie_iE_i\), he shows that not only is it possible to avoid the mixed case, rather, when the mmp terminates in \(X^m\) and \(K_{X^m}+\Theta^m\) is effective, one obtains \(X^{m} = Y\) instead of an unknown small modification of \(Y\). Furthermore, when \(-(K_{X^m}+\Theta^m)\) is effective, \(g_m\colon X^m\to Y\) contracts nothing extra to \(g_m(\operatorname{Supp}(K_{X^m}+\Theta^m))\). This result allows the author to remove the \(\mathbb{Q}\)-factorial assumption in many well-known results. For example, he shows that when \((Y, \Delta)\) and \((X, E+g_*^{-1}\Delta)\) are dlt pairs in characteristic 0, the discrepancies of \(E_i\) are greater that \(-1\), and \(E\) supports a relative ample divisor, then the dual complex of \(E_y\), the divisorial part of \(g^{-1}(y)\), is either empty or is collapsible to a point. Other results from which the author removed the \(\mathbb{Q}\)-factoriality assumptions are the inversion of adjunction formula for 3-folds in low characteristics, Grauert--Riemenschneider vanishing etc.