7 - Conical Fibrations

Summary

A Mori fibre space of relative dimension one is a conical fibration and it is birational to a standard conic bundle. The object in this chapter is a Q-conic bundle, namely a Mori fibre space from a threefold to a surface. The analytic germ along a fibre is analogous to an extremal neighbourhood but the higher direct image of the canonical sheaf may not vanish. Mori and Prokhorov achieved a classification of Q-conic bundle germs and the general elephant conjecture when the central fibre is irreducible. This implies Iskovskikh's conjecture that the base surface is Du Val of type A. We also discuss the rationality problem of threefold standard conic bundles over a rational surface. Beauville, after Mumford, proved that the intermediate Jacobian is isomorphic to the Prym variety of the double cover of the discriminant divisor. By virtue of Shokurov's analysis, we have a complete criterion for rationality when the base is relatively minimal. Sarkisov proved that a standard conic bundle is birationally superrigid if a certain pseudo-effective threshold is at least four. In contrast, the rationality in dimension three is nearly paraphrased as having the threshold less than two.