8 - Del Pezzo Fibrations

Summary

A Mori fibre space of relative dimension two is a del Pezzo fibration. We address the problem of finding a good representative in each equivalence class of threefold del Pezzo fibrations. In the case of degree at least three, Corti realised a standard model embedded anti-canonically into a projective space bundle. We explain the construction by Kollár for degree three as a semistable model. The stability theory completes the existence of standard models for degree two and one. Mori and Prokhorov proved that the multiplicity of a fibre is bounded by six. We also discuss the rationality problem of del Pezzo fibrations to a rational curve. The total space is always rational when the degree is at least five. We exhibit Alexeev's work for degree four. He established a criterion for rationality in terms of topological Euler characteristic. The lower degree case is treated from the point of view of birational rigidity. Pukhlikov proposed the K2-condition and proved that it is sufficient for birational rigidity. We compare it with Grinenko's K-condition. For a fibration of degree one from a smooth threefold, the K-condition turns out to be equivalent to the birational superrigidity.