On Fibrations into del Pezzo Surfaces

The problem of birational rigidity for three-dimensional algebraic varieties fibered over rational curves into del Pezzo surfaces of degree 1 is discussed. A criterion for the rigidity of such fibrations in the Mori category is suggested and the inverse implication is proved (Theorem 3.3). Surgeries on fibers in fibrations of this type, which turn out to be closely related to the rigidity problem, are considered. In particular, an important result on the uniqueness of a smooth model in a class of maps over a base is stated (Corollary 4.5).