Book contents
- Frontmatter
- Contents
- List of Contributors
- Preface
- 1 Birational Geometry of Algebraic Varieties and Shokurov’s Work
- 2 ACC for Log Canonical Thresholds for Complex Analytic Spaces
- 3 Conjectures on the Kodaira Dimension
- 4 Characterizing Terminal Fano Threefolds with the Smallest Anti-Canonical Volume, II
- 5 Uniform Rational Polytopes for Iitaka Dimensions
- 6 MMP for Algebraically Integrable Foliations
- 7 On Toric Fano Fibrations
- 8 Q-Fano Threefolds of Fano Index 13
- 9 Reflective Hyperbolic 2-Elementary Lattices, K3 Surfaces and Hyperkahler Manifolds
- 10 The Relative Du Bois Complex – on a Question of S. Zucker
- 11 Factorization Presentations
- 12 Spectrum Bounds in Geometry
- 13 On the DCC of Iitaka Volumes
- 14 Shokurov’s Index Conjecture for Quotient Singularities
- 15 A Note on the Sarkisov Program
- 16 Cluster Varieties and Toric Specializations of Fano Varieties
- 17 Birational Rigidity and Alpha Invariants of Fano Varieties
- 18 On F-Pure Inversion of Adjunction
- 19 On Termination of Flips and Fundamental Groups
- 20 Motivic Integration on Berkovich Spaces
16 - Cluster Varieties and Toric Specializations of Fano Varieties
Published online by Cambridge University Press: 06 December 2024
- Frontmatter
- Contents
- List of Contributors
- Preface
- 1 Birational Geometry of Algebraic Varieties and Shokurov’s Work
- 2 ACC for Log Canonical Thresholds for Complex Analytic Spaces
- 3 Conjectures on the Kodaira Dimension
- 4 Characterizing Terminal Fano Threefolds with the Smallest Anti-Canonical Volume, II
- 5 Uniform Rational Polytopes for Iitaka Dimensions
- 6 MMP for Algebraically Integrable Foliations
- 7 On Toric Fano Fibrations
- 8 Q-Fano Threefolds of Fano Index 13
- 9 Reflective Hyperbolic 2-Elementary Lattices, K3 Surfaces and Hyperkahler Manifolds
- 10 The Relative Du Bois Complex – on a Question of S. Zucker
- 11 Factorization Presentations
- 12 Spectrum Bounds in Geometry
- 13 On the DCC of Iitaka Volumes
- 14 Shokurov’s Index Conjecture for Quotient Singularities
- 15 A Note on the Sarkisov Program
- 16 Cluster Varieties and Toric Specializations of Fano Varieties
- 17 Birational Rigidity and Alpha Invariants of Fano Varieties
- 18 On F-Pure Inversion of Adjunction
- 19 On Termination of Flips and Fundamental Groups
- 20 Motivic Integration on Berkovich Spaces
Summary
I state a conjecture describing the set of toric specializations of a Fano variety with klt singularities. The conjecture asserts that for all generic Fano varieties X with klt singularities, there exists a polarized cluster variety U and a surjection from the set of torus charts on U to the set of toric specializations of X.
I outline the first steps of a theory of the cluster varieties that I use. In dimension 2, I sketch a proof of the conjecture after Kasprzyk–Nill–Prince, Lutz, and Hacking by way of work of Lai–Zhou. This reveals a surprising structure to the classification of log del Pezzo surfaces that was first conjectured in [1]. In higher dimensions, I survey the evidence from the Fanosearch program, cluster structures for Grassmannians and flag varieties, and moduli spaces of conformal blocks.
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- Chapter
- Information
- Higher Dimensional Algebraic GeometryA Volume in Honor of V. V. Shokurov, pp. 264 - 285Publisher: Cambridge University PressPrint publication year: 2025