On the flip theorem for locally primitive extremal rays in dimension 3

Abstract.

The aim of this paper is to give an other proof of Mori's [Mo1] famous existence of flip for 3-folds in the case of extremal rays on 3-fold which are locally primitive. The proof starts with some original arguments from [Mo1] and reduces the proof to the case of index 1 terminal singularities. Nevertheless it does not use Kawamata's argument involving double canonical covers which is the method Mori used. Instead it uses the blowing up of the completion of the 3-fold along the curve defining an extremal ray.