Fano varieties in Mori fibre spaces

TL;DR

This paper investigates the conditions under which Fano varieties can be fibers in Mori fibre spaces, providing criteria, characterizations, and applications across various classes including toric and rational homogeneous spaces.

Contribution

It introduces new necessary and sufficient criteria for Fano varieties to be fibers of Mori fibre spaces, advancing understanding in algebraic geometry.

Findings

Criteria for Fano fibers in Mori spaces established

Characterization in the rigid case achieved

Connections with K-semistability explored

Abstract

We show that being a general fibre of a Mori fibre space is a rather restrictive condition for a Fano variety. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal singularities to be realised as a fibre of a Mori fibre space, which turn into a characterisation in the rigid case. We apply our criteria to figure out this property up to dimension three and on rational homogeneous spaces. The smooth toric case is studied and an interesting connection with K-semistability is also investigated.