On families of rational curves and Fano varieties.

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The reporter studied under what conditions a surjective morphism f: X -> Y is locally trivial or isomorphic. It is important and effective toclassify the world of Fano varieties. In the formar case assuming that the anti-relative canonical line bundle -K_f (:= -K_X -f*K_Y) of the morphism f is nef,we can show f is locally trivial. Especially in case of the smoothness of f assuming that a general fiber is Del pezzo surface or a smooth hypersurface,we get the same results. In the latter case if X is an n(> 2)-dimensional cubic hypersurface, f is an isomorphism unless Y is a projective space or a hyperquadric.