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§ O. Introduction § I. Preliminaries and basic definitions § la (Appendix la). Results on 3-fold terminal singularities § Ib (Appendix Ib). Deformation of extremal nbds § 2. Numerical invariants ip(n), wp(O), and w;(n) § 3. Embedding dimension of (en, pn) § 4. Classification of X ::> e at P into cases § 5. Numerical calculations for (IA), (IC), (IA v) , and (IC v) § 6. Possible singularities on an extremal nbd X::> e === Wi § 7. Existence of "good" anti(bi)canonical divisor (easy case) § 8. J -filtration and lifting structure § 9. Existence of "good" anticanonical divisor (delicate case) §IO. A slight generalization of (0.4.5) and comments on (la.7) Appendix A. Summary of local classification of X::> e 3 P Appendix B. Possible singularities on an extremal nbd List of notation and terminology References

Flip theorem and the existence of minimal models for 3-folds