Affine threefolds whose log canonical bundles are not numerically effective

Abstract: Let be a compactification of an affine 3-fold into a smooth projective 3-fold such that the (reduced) boundary divisor 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 -negative extremal rays on . In particular, if such an extremal ray intersects non-negatively, we shall describe the log flips and divisorial contractions appearing explicitly. [Copyright &y& Elsevier]