Affine threefolds whose log canonical bundles are not numerically effective

Let Xright arrow-hooked(T,D) be a compactification of an affine 3-fold X into a smooth projective 3-fold T such that the (reduced) boundary divisor D is SNC. In this paper, as an affine counterpart to the work due to S. Mori (cf. [S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116 (1982) 133–176]), we shall classify (K+D)-negative extremal rays on T. In particular, if such an extremal ray image intersects K non-negatively, we shall describe the log flips and divisorial contractions appearing explicitly.