Automorphisms of ℂ3Commuting with a ℂ+-Action

2014/06/30 by Immanuel Stampfli
Mathematics · #Action (physics) #Advanced Differential Equations and Dynamical Systems #Affine transformation #Algebraic Geometry and Number Theory #Algebraic number #Automorphism #Automorphisms of the symmetric and alternating groups #Geometric and Algebraic Topology #Group (periodic table) #Identity (music) #Polynomial #Quotient #math.AG

paper · pdf · doi:10.1093/imrn/rnu232

published as Int Math Res Notices (2015) 2015 (19): 9832-9856 · 18 pages, typos corrected, minor changes, improved exposition

openalex publication_date 2014/12/16 · openalex created_date 2016/06/24 · arxiv created 2016/11/23 · arxiv updated 2016/11/24 · openalex updated_date 2026/08/05

Abstract

Let |ρ| be an algebraic action of the additive group |\mathbb C⁺ | on the three-dimensional affine space |\mathbb C³|⁠. We describe the group |\mathrm Cent(ρ )| of polynomial automorphisms of |\mathbb C³| that commute with |ρ|⁠. A particular emphasis lies in the description of the automorphisms in |\mathrm Cent(ρ )| coming from algebraic |\mathbb C⁺ |-actions. As an application, we prove that the automorphisms in |\mathrm Cent(ρ )| that are the identity on the algebraic quotient of |ρ| form a characteristic subgroup of |\mathrm Cent(ρ )|⁠.

Citations

• Automorphisms of the plane preserving a curve
• On Automorphisms of the Affine Cremona Group
• Invariants and separating morphisms for algebraic group actions
• Dynamical properties of plane polynomial automorphisms
• A note on automorphisms of the affine Cremona group