Geodesic
A rationality criterion for conic bundles
Sbornik. Mathematics, Tome 187 (1996) no. 7, pp. 1021-1038

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It is that a three-dimensional variety $X$ that is a conic bundle $\pi\colon X\to S$ in the Mori sense has a base with at most double rational singularities of type $A_n$. A rationality criterion is proved subject to this assumption in the case when the discriminant curve $C\subset S$ is large enough, for example, for the case when $p_a(C)>18$. 
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     author = {V. A. Iskovskikh},
     title = {A rationality criterion for conic bundles},
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V. A. Iskovskikh. A rationality criterion for conic bundles. Sbornik. Mathematics, Tome 187 (1996) no. 7, pp. 1021-1038. http://geodesic.mathdoc.fr/item/SM_1996_187_7_a3/