On Contraction Of Extremal Face Which Is Mori Fibration

Let X be a non-singular variety of dimension n, L be an ample linebundle over X andK Xbe the canonical line bundle over X. Let f:X→Ybe acontraction morphism of extremal face from X to Y with supporting divisorK_X+(n-4)L. Suppose that X is not birational equivalent to Y,(X, L)must bea class of special varieties. The complete classification of (X, L)is obtainedas following:(i)(X, L)is a Fano variety of index n4;(ii)(X, L)is a Fano fibration of co-index4over a smooth curve;(iii)(X, L)is a Mukai fibration over a normal surface;(iv)(X, L)is a Del Pezzo fibration over a normal3-dimensional variety;(v)(X, L)is a quadric fibration over a normal4-dimensional variety;(vi)(X, L)is a scroll over a normal5-dimensional variety.