On the rationality problem for conic bundles
The author posed the following two conjectures for the rationality criterion of conic bundles in dimension three: Conjecture I. Let \(\pi:V\to S\) be a standard conic bundle on a rational surface \(S\) with connected discriminant curve \(C\). Then \(V\) is rational if and only if one of the following two conditions are satisfied: (i) \(\pi\) is birationally equivalent to a standard conic bundle \(\pi': V'\to '\) with discriminant C' such that S' has a fibration \(S'\to {\mathbb{P}}^ 1\) with \({\mathbb{P}}^ 1\) as general fibers and that \(\deg(C'\to {\mathbb{P}}^ 1)\leq 3\). (ii) \(\pi\) is birationally equivalent to a standard conic bundle \(\pi_ 0: V_ 0\to {\mathbb{P}}^ 2\) with discriminant \(C_ 0\subseteq {\mathbb{P}}^ 2\) of degree 5 such that the associated double covering \(\tilde C_ 0\to C_ 0\) is defined by an even theta- characteristic. Conjecture II. For any rational standard conic bundle \(\pi:V\to S,\) there is a linear system \(\Lambda \subseteq | \pi^*D-K_ V| (D\in Pic(S))\) such that \(\Phi_{\Lambda}: V\to {\mathbb{P}}^ 3\) is birational and \(\Phi_{\Lambda}(\pi^{-1}(S))\) are conics for \(s\in S.\) The author proves that the above conjectures are equivalent to each other in {\S} 1. - In {\S} 2, he tries to prove conjecture I by considering a different conic bundle structure on V and poses another conjecture III.
scientific article; zbMATH DE number 4023476
scientific article