Cluster varieties and toric specializations of Fano varieties
File(s)
Accepted version
Author(s)
Corti, Alessio
Type
Conference Paper
Abstract
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.
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.
Date Acceptance
2023-11-02
Citation
London Mathematical Society Lecture Note Series
ISSN
0076-0552
Publisher
Cambridge University Press
Journal / Book Title
London Mathematical Society Lecture Note Series
Copyright Statement
Subject to copyright.
Source
JAMI Conference 2022: Higher Dimensional Algebraic Geometry
Publication Status
Accepted
Start Date
2022-05-03
Finish Date
2022-05-08
Coverage Spatial
Baltimore, MD, USA
Rights Embargo Date
10000-01-01