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 $\mathbb P_{\mathbb P^1}(\mathscr O(d_1)\oplus\dots\oplus\mathscr O(d_5))$ is obtained. The following addition to Alexeev's theorem on the rationality of standard Del Pezzo fibrations of degree 4 over $\mathbb P^1$ is deduced as a consequence: each fibration of this kind with topological Euler characteristic $\chi(X)=-4$ is proved to be rational.
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