Fano Varieties in Mori Fibre Spaces

2014/06/30 by Giulio Codogni, Andrea Fanelli, Roberto Svaldi +1
Mathematics · #Algebraic Geometry and Number Theory #Combinatorics #Computer science #Connection (principal bundle) #Dimension (graph theory) #Factorial #Fano plane #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Gravitational singularity #Homogeneous #Mathematical analysis #Mathematics #Pure mathematics #Space (punctuation) #Variety (cybernetics) #math.AG #msc:14E30 #msc:14J26 #msc:14J30 #msc:14J45 #msc:14M15 #msc:14M25

paper · pdf · doi:10.1093/imrn/rnv173

published as Int Math Res Notices (2016) 2016 (7): 2026-2067 · 32 pages. Results on threefolds and rational homogeneous spaces strengthened. Result on toric varieties corrected. To appear in IMRN

arxiv created 2015/05/20 · openalex publication_date 2015/06/25 · arxiv updated 2016/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We show that being a general fibre of a Mori fibre space (MFS) is a rather restrictive condition for a Fano variety. More specifically, we obtain two criteria (one sufficient and one necessary) for a |\mathbb 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 3 and on rational homogeneous spaces. The smooth toric case is studied and an interesting connection with |K|-semistability is also investigated.

Citations

• Principles of Algebraic Geometry
• Birational Geometry of Algebraic Varieties
• Positivity in Algebraic Geometry I
• Birational Geometry of 3-fold Mori Fibre Spaces
• Classification of Fano 3-folds with B 2 ?2
• Stratifications de Whitney et théorème de Bertini-Sard
• Existence of minimal models for varieties of log general type
• Classification of Fano 3-folds with B2?2
• Flip Theorem and the Existence of Minimal Models for 3-Folds
• Compact Complex Surfaces