Algebraic threefolds with two extremal morphisms

Abstract

In [3] Mori gives a description of all extremal rays (extremal morphisms) arising on a smooth projective threefold with a numerically non-effective canonical bundle. Generally speaking, every smooth projective threefold V with a numerically non-effective canonical class Kv admits an extremal morphism π: V— → Y. The assumption that V admits a non-trivial pair of extremal morph-ismsimposes strong conditions on V. This is the essence of the Theorem 1.5 of the present work. In particular, we obtain a description of the threefolds which admit two biregular structures of conic bundles over non-singular surfaces S1 = Y1 and S2 = Y2. By the results of §3 the surfaces Sl and S2 must be either ruled surfaces with isomorphic basic curves, or S1 ≈ S2 ≈ P2.