Geodesic
On the rationality of non-singular threefolds with a pencil of Del~Pezzo surfaces of degree~4
Sbornik. Mathematics, Tome 197 (2006) no. 1, pp. 127-137

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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. Bibliography: 10 titles. 
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     author = {K. A. Shramov},
     title = {On the rationality of non-singular threefolds with a pencil of {Del~Pezzo} surfaces of degree~4},
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K. A. Shramov. On the rationality of non-singular threefolds with a pencil of Del~Pezzo surfaces of degree~4. Sbornik. Mathematics, Tome 197 (2006) no. 1, pp. 127-137. http://geodesic.mathdoc.fr/item/SM_2006_197_1_a6/