On the rationality of non-singular threefolds with a pencil of Del Pezzo surfaces of degree 4

A criterion for the non-singularity of a complete intersection of two fibrewise quadrics in PP1(O(d1)+...+O(d5)) is obtained. The following addition to Alexeev's theorem on the rationality of standard Del Pezzo fibrations of degree 4 over P1 is deduced as a consequence: each fibration of this kind with topological Euler characteristic χ(X)=-4 is proved to be rational.