Nonrational Del Pezzo Fibrations

In this paper we study cubic del Pezzo fibrations f : X → P such that the 3-fold X is smooth and rkPic(X) = 2. These are the examples of smooth 3-fold Mori fibre spaces. The 3-fold X is a divisor in the linear system |3M + nL| on the rational scroll Proj(⊕ i=1 OP1(di)), where n and di are integers, d1 ≥ d2 ≥ d3 ≥ d4 = 0, M is a tautological line bundle, and L is a fibre of the projection to P. The 3-fold X is rational in the case d1 = d2 = d3 = d4 = 0 and n = 1, i.e. when X is a divisor of bi-degree (1, 3) on P × P. In all other cases we prove the nonrationality of X in the additional assumption that the 3-fold X is a sufficiently general divisor in the linear system |3M + nL|. As a corollary we prove that every smooth 3-fold with Picard rank 2 fibred into del Pezzo surfaces of degree less or equal than 3 is a smooth deformation of nonrational smooth 3-folds except the case of a divisor of bi-degree (1, 3) on P × P.