2017/11/28 by N. I. Shepherd‐Barron, Shepherd-Barron, N. I. · 1 citation
Mathematics · #13A50 #14B07 #14J17 #14M15 #57T15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds
paper · doi:10.48550/arxiv.1711.10439
openalex publication_date 2017/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We unify results of Artin, Brieskorn, Slodowy and others by showing that, in all characteristics, the Artin component of the deformation space of a rational surface singularity has a ramified cover where simultaneous resolution exists and the Galois group of this cover is the Weyl group determined by the configuration of (-2)-curves in the minimal resolution. This verifies a conjecture made by Burns and Rapoport. We use an integral version of this to show that certain actions of Weyl groups on polynomial rings over the integers give rings of invariants that are also polynomial, and deduce that the integral cohomology rings of complete flag varieties G/B of types A,D or E can be described as the corresponding rings of co-invariants.