Birationally Rigid Del Pezzo Fibrations

In this paper we study del Pezzo fibrations z : X → P1 of degree 1 and 2 such that X is smooth, rk Pic(X) = 2 and K2 X ∈ NE(X). These are examples of smooth birationally rigid 3-fold Mori fibre spaces. We describe all birational transformations of the 3-fold X into elliptic fibrations, fibrations of surfaces of Kodaira dimension zero, and canonical Fano 3-folds. Let X be a smooth 3-fold1 of Picard rank 2 and τ : X → P1 be a flat morphism whose generic fiber is a del Pezzo surface of degree 1 or 2. The following result is proved in [18]. Theorem 1. Suppose that K2 X ∈ Int(NE(X)). Then X is not birational to the following: • a conic bundle; • a fibration2 of rational surfaces that is not equivalent 3 to τ ; • a Fano 3-fold of Picard rank 1 having terminal Q-factorial singularities. The main purpose of this paper is to prove the following result. Theorem 2. Suppose that K2 X ∈ NE(X). Then X is birational neither to a fibration whose generic fiber is a surface of Kodaira dimension zero nor to a Fano 3-fold I. Cheltsov: Steklov Institute of Mathematics 8 Gubkin street, Moscow, 117966 Russia. e-mail: [email protected] I. Cheltsov: School of Mathematics, University of Edinburgh, Kings Buildings, Mayfield Road Edinburgh EH9 3JZ, UK. e-mail: [email protected] 1 All varieties are assumed to be projective, normal, and defined over the field C. 2 For every fibrationπ :Y →Z we assume that dim(Y )>dim(Z) =0 andπ∗(OY )=OZ . 3 Fibrations τ : U → Z and τ̄ : Ū → Z̄ are called equivalent if there are birational maps α : U Ū and β : Z Z̄ such that the diagram