Affine threefolds whose log canonical bundles are not numerically effective

Let X↪(T,D)X↪(T,D) be a compactification of an affine 3-fold XX into a smooth projective 3-fold TT such that the (reduced) boundary divisor DD 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)(K+D)-negative extremal rays on TT. In particular, if such an extremal ray R=R+[C]R=R+[C] intersects KK non-negatively, we shall describe the log flips and divisorial contractions appearing explicitly.