2025/10/08 by Simon Crawford, Crawford, Simon, Susan J. Sierra +1
Mathematics · #14E15. Secondary: 16S38 #14E16 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Primary: 14A22 #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2510.07137
openalex publication_date 2025/10/08 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
Let B = \Bbbkq[u,v]^Cn+1 be a Type \mathbbAn quantum Kleinian singularity, which is an example of a noncommutative surface singularity. This singularity is known to have a noncommutative quasi-crepant resolution Λ, which is an "algebraic" resolution of B. We construct a category X which serves as a "geometric" resolution of B by adapting techniques from quiver GIT and show that X and mod-Λ are derived equivalent. Furthermore, we show that the intersection arrangement of lines in the exceptional locus of X corresponds to a Type \mathbbAn Dynkin diagram. This generalises the geometric McKay correspondence for classical Kleinian singularities.