CANONICAL FACTORIZATION OF THE QUOTIENT MORPHISM FOR AN AFFINE G $$ \mathbb{G} $$ a-VARIETY

Abstract

Working over a ground field of characteristic zero, this paper studies the quotient morphism π : X → Y for an affine $$ \mathbb{G} $$ a-variety X with affine quotient Y. It is shown that the degree modules associated to the Ga-action give a uniquely determined sequence of dominant $$ \mathbb{G} $$ a-equivariant morphisms, $$ X={X}_r\to {X}_{r-1}\to \cdots \to {X}_1\to {X}_0=Y, $$ where Xi is an affine $$ \mathbb{G} $$ a-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 $$ \mathbb{G} $$ a-action, and this yields the canonical factorization of 𝜋. The algorithm is applied to compute the canonical factorization for several examples, including the homogeneous (2, 5) action on $$ {\mathbb{A}}^3 $$ The Freeness Conjecture, introduced in the paper's last section, asserts that, for any $$ \mathbb{G} $$ a-action on $$ X={\mathbb{A}}^3 $$ the polynomial ring k[X] is a free module over $$ k{\left[X\right]}^{{\mathbb{G}}_a} $$ .