Affine threefolds whose log canonical bundles are not numerically effective (Q856344)

Let \(X\) be a smooth affine complex threefold embedded into a smooth projective threefold \(T\) such that the reduced boundary divisor \(D\) has only simple normal crossings and the divisor \(K_T+D\) is not nef. The author is interested in the birational geometry of \(X\) and applies the \((K_T+D)\)-minimal model program \[ (T,D){\overset\varphi{^0}\dashrightarrow}(T^1,D^1)\ldots\dashrightarrow \dots{\overset\varphi^{s-1}\dashrightarrow}(T^s,D^s) \] to \((T,D)\). To keep the data of \(X\) under control in this process it is necessary to get a precise analysis of every step \(\varphi^i:(T^i,D^i)\dashrightarrow(T^{i+1},D^{i+1})\). The author gives an explicit description of the first step \(\phi^0\), namely a classification of the \((K_T+D)\)-negative extremal rays \(\mathbb R_+[C]\) on \(T\) and considers this result as basic for an inductive description of the following steps \(\phi^i\). The classification is carried out in a case-by-case investigation, based on the methods and results of the minimal model program. The main part of the classification is related to the situation \((K_T\cdot C)\geq 0\), when \(\log\) flips may occur.