Birational Properties of Pencils of Del Pezzo Surfaces of Degree 1 and 2

In this paper we study the birational rigidity problem for smooth Mori fibrations on del Pezzo surfaces of degree 1 and 2. For degree 1 we obtain a complete description of rigid and non-rigid cases. 1. Notions and results. Due to progress in the Minimal Model Program, the classification problem in the modern birational geometry for varieties of negative Kodaira dimension can be formulated as follows: given a class of birational equivalency, to describe all Mori fibrations and birational maps between them. Recall that a normal variety V with at most Q-factorial terminal singularities is called a Mori fibration if there exists an extremal contraction φ : V → S of fibering type, i.e. 1) φ is a morphism with connected fibers onto a normal variety S and dimS < dim V ; 2) −KV is φ-ample and the relative Picard number is equal to 1: ρ(V/S) = ρ(V )− ρ(S) = 1. Note that often comparing birational classes of varieties, we suffice to know Mori structures rather than Mori fibrations themselves (roughly speaking, Mori fibrations modulo birational maps over the base). From this viewpoint, the following class of varieties is rather important (and simple for describing): Definition 1.1. A Mori fibration V/S is said to be birationally rigid, if any birational map χ : V 99K V ′ onto another Mori fibration V /S ′ is birational over the base (”square”), i.e., there exists a birational map This work was partially supported by the grants RFBR no. 99–01–01132, Grant of Leading Scientific Schools no. 96–15–96146, and INTAS-OPEN 97/2072.