Some contractions of threefold algebraic family (Q1333959)

By the central result of the Mori theory, the possible extremal morphisms on a smooth projective threefold can be distinguished from each other by the numerical properties of their corresponding extremal curves and supporting divisors. In this paper the author studies special cases of threefolds of negative Kodaira dimension and with small Picard number, and finds which of the Mori extremal contractions are possible for these threefolds (theorems 2.2 and 2.3). The study is based on the known numerical properties of possible extremal curves, and on the generalization of the rationality theorem proved in the beginning (theorem 2.1), which imposes additional numerical restrictions on these extremal curves.