Canonical factorization of the quotient morphism for an affine $\mathbb{G}_a$-variety

Abstract

Working over a ground field of characteristic zero, this paper studies the quotient morphism $\pi :X\to Y$ for an affine $\mathbb{G}_a$-variety $X$ with affine quotient $Y$. It is shown that the degree modules associated to the $\mathbb{G}_a$-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 $X_i$ is an affine $\mathbb{G}_a$-variety and $X_{i+1}\to X_i$ is birational for each $i\ge 1$. This is the canonical factorization of $\pi$. We give an algorithm for finding the degree modules associated to the given $\mathbb{G}_a$-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 $\mathbb{A}^3$. 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 $\mathbb{G}_a$-equivariant affine modification.