Congruences of conics in \({\mathbb{P}}^ 3\) (Q788790)

The author studies the problem of rationality of an algebraic threefold V admitting a surjective morphism to a smooth projective surface S whose fibres are isomorphic to conics (a conic bundle). A condition of rationality of such a threefold is given in terms of the discriminant curve \(C\subset S\) of this morphism. Namely, V is rational if \(C=\emptyset\), or there exists a pencil of rational curves on S inducing a rational map \(C\to {\mathbb{P}}^ 1\) of degree at most 3, or there exists a birational map \(S\to {\mathbb{P}}^ 2\) which sends C to a curve of degree at most 5 and, in the case of the degree equal to 5, the double cover of C induced by the conic bundle structure corresponds to an even theta characteristic (the latter condition was essentially known earlier). The author points out that the above conditions are necessary provided the following fact is true: Every congruence of index 1 of rational curves in \({\mathbb{P}}^ 3\) can be reduced by Cremona transformations to a congruence of conics.