Nonexistence of degree two rational multisections of conic bundles over the plane

Abstract

We prove that a standard conic bundle $X\to {\mathbb{P}}_{\mathbb{C}}^2$ whose discriminant is very general of degree at least 18 admits no rational multisections of degree two. This is the first step towards proving a conjecture of Iskovskikh that there are conic bundle threefolds that are not unirational, since to prove that $X$ is not unirational, it suffices to show that there are no rational multisections of any degree. Proving Iskovskikh's conjecture would provide the first example of a rationally connected variety that is not unirational.

10 pages