Log models of birationally rigid varieties
Let \(X\) be a projective manifold \(X\) of dimension \(n\geq 2\). From the point of view of the Mori theory one of the important points in understanding the birational geometry of \(X\) is the problem of describing all the Mori spaces which are birationally isomorphic to \(X\). Recall that a Mori space is a morphism \(f:Z\to S\), where \(Z\) is a projective variety having at most terminal \(\mathbb Q\)-factorial singularities, with \(f\)-ample anticanonical class and one-dimensional relative Picard group, and \(S\) is a normal projective variety of dimension smaller than the dimension of \(Z\) such that \(f_*(\mathcal O_Z)=\mathcal O_S\). In the paper under review the author uses the concept of movable log pairs \((X,M)\) (where \(M=\sum_ib_iM_i\), with \(b_i\) nonnegative rational numbers and \(M_i\) linear systems on \(X\) without fixed components) to study this question, in fact to classify all log models of a given projective manifold \(X\). His idea is to translate the existence of log pairs into numerical constraint of the linear system. Some of the results obtained generalize known results for \(3\)-folds.
scientific article; zbMATH DE number 1551881
scientific article