Log canonical pairs with good augmented base~loci (Q2877492)

The authors show that if \((X,B)\) is a projective log canonical pair, \(B\) is a \(\mathbb Q\)-divisor, \(f:X\to Z\) is a surjective morphism of normal varieties, \(K_X+B\sim _{\mathbb Q}f^*M\) where \(M\) is a big \(\mathbb Q\)-divisor on \(Z\), and the augmented stable base locus \({\mathbf B} _+(M)\) does not contain the image of any log canonical center of \((X,B)\) then \((X,B)\) has a good log minimal model. As a consequence the authors also show that if \((X,B)\) is a projective log canonical pair, \(K_X+B\) is a big \(\mathbb Q\)-divisor and \({\mathbf B} _+(K_X+B)\) does not contain log canonical centers of \((X,B)\), then \((X,B)\) has a good minimal model and hence the log canonical ring \(R(K_X+B)\) is finitely generated.