Fano varieties in Mori fibre spaces

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.

32 pages. Results on threefolds and rational homogeneous spaces strengthened. Result on toric varieties corrected. To appear in IMRN

References in corpus (5)

• On G-Fano threefolds
• Quantum Periods for 3-Dimensional Fano Manifolds
• An algorithm for the classification of smooth Fano polytopes
• Birational Geometry of 3-fold Mori Fibre Spaces
• Rigidity properties of Fano varieties

Cited by in corpus (9)

• On K-stability of finite covers
• Connected algebraic groups acting on three-dimensional Mori fibrations
• Examples of K-unstable Fano manifolds with the Picard number one
• A note on the fibres of Mori fibre spaces
• Rational curves on $V_5$ and rational simple connectedness
• Fano 3-folds from homogeneous vector bundles over Grassmannians
• Fano 4-folds having a prime divisor of Picard number 1
• Balanced line bundles on Fano varieties
• Normal split divisors in rational homogeneous spaces