Flops and mutations for crepant resolutions of polyhedral singularities

2011/08/11 by Álvaro Nolla de Celis, de Celis, Alvaro Nolla, Yuhi Sekiya +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1108.2352

openalex publication_date 2011/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a polyhedral group G⊂ SO(3) of types ℤ/nℤ, D_2n and \mathbbT. We prove that there exists a one-to-one correspondence between flops of G-Hilbℂ³ and mutations of the McKay quiver with potential which do not mutate the trivial vertex. This correspondence provides two equivalent methods to construct every projective crepant resolution for the singularities ℂ³/G, which are constructed as moduli spaces M_C of quivers with potential for some chamber C in the space Θ of stability conditions. In addition, we study the relation between the exceptional locus in M_C with the corresponding quiver Q_C, and we describe explicitly the part of the chamber structure in Θ where every such resolution can be found.

Related

• Crepant resolutions, mutations, and the space of potentials
• Crepant resolutions of orbit closures in Quiver representations of type A_n and D₄
• Mutations of quivers with 2-cycles
• Resolutions of Orbifold Singularities and Flows on the McKay Quiver
• Quiver braid group action for a 3-fold crepant resolution