Geodesic
Birational automorphisms of a~class of varieties fibred into cubic surfaces
Izvestiya. Mathematics , Tome 66 (2002) no. 1, pp. 201-222.

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We investigate the properties of a variety $V$ that is a divisor of bidegree $(2,3)$ in $\mathbb P^1\times\mathbb P^3$. We calculate the group of its birational automorphisms and prove that $V$ admits no conic bundle structure and is not rational. 
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I. V. Sobolev. Birational automorphisms of a~class of varieties fibred into cubic surfaces. Izvestiya. Mathematics , Tome 66 (2002) no. 1, pp. 201-222. http://geodesic.mathdoc.fr/item/IM2_2002_66_1_a8/

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