A note on the fibres of Mori fibre spaces

TL;DR

This paper investigates which Fano manifolds can serve as fibers in Mori fiber spaces, focusing on toric varieties, high index Fano manifolds, and those with high Picard rank, to understand their structural roles.

Contribution

It provides new insights into the classification of Fano manifolds as fibers of Mori fiber spaces, especially in specific classes like toric and high-index varieties.

Findings

Identifies conditions under which certain Fano manifolds appear as Mori fiber space fibers.

Highlights the role of toric and high Picard rank Fano manifolds in this context.

Provides partial classifications and examples for these cases.

Abstract

In this note we consider the problem of determining which Fano manifolds can be realised as fibres of a Mori fibre space. In particular, we study the case of toric varieties, Fano manifolds with high index and some Fano manifolds with high Picard rank.