2016/02/28 by Gene Freudenburg, Freudenburg, Gene
Mathematics · #14R10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1602.08786
openalex publication_date 2016/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Working over a ground field of characteristic zero, this paper studies the quotient morphism π:X→ Y for an affine \mathbbGa-variety X with affine quotient Y. It is shown that the degree modules associated to the \mathbbGa-action give a uniquely determined sequence of dominant \mathbbGa-equivariant morphisms, X=Xr→ Xr-1→⋯→ X1→ X0=Y, where Xi is an affine \mathbbGa-variety and Xi+1→ Xi is birational for each i≥ 1. This is the canonical factorization of π. We give an algorithm for finding the degree modules associated to the given \mathbbGa-action, and this yields the canonical factorization of the quotient morphism. The algorithm is applied to compute the canonical factorization for several examples, including the homogeneous (2,5)-action on \mathbbA3. By a fundamental result of Kaliman and Zaidenberg, any birational morphism of affine varieties is an affine modification, and each mapping in these examples is presented as a \mathbbGa-equivariant affine modification.