Mori contractions of maximal length

TL;DR

This paper proves that Mori fiber spaces of maximal length are birationally equivalent to projective bundles, extending a key theorem in algebraic geometry to a relative setting.

Contribution

It provides a relative version of the theorem by Cho, Miyaoka, and Shepherd-Barron, establishing a new birational classification result for Mori fiber spaces.

Findings

Mori fiber spaces of maximal length are birational to projective bundles.

Extension of the theorem of Cho, Miyaoka, and Shepherd-Barron to a relative context.

Advances understanding of the structure of Mori contractions.

Abstract

We prove a relative version of the theorem of Cho, Miyaoka and Shepherd-Barron: a Mori fibre space of maximal length is birational to a projective bundle.