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Working over a ground field of characteristic zero, this paper studies thequotient morphism $\pi :X\to Y$ for an affine $\mathbb{G}_a$-variety $X$ withaffine 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, $X=X_r\to X_{r-1}\to\cdots\to X_1\toX_0=Y$, where $X_i$ is an affine $\mathbb{G}_a$-variety and $X_{i+1}\to X_i$ isbirational for each $i\ge 1$. This is the canonical factorization of $\pi$. Wegive an algorithm for finding the degree modules associated to the given$\mathbb{G}_a$-action, and this yields the canonical factorization of thequotient morphism. The algorithm is applied to compute the canonicalfactorization for several examples, including the homogeneous $(2,5)$-action on$\mathbb{A}^3$. By a fundamental result of Kaliman and Zaidenberg, anybirational morphism of affine varieties is an affine modification, and eachmapping in these examples is presented as a $\mathbb{G}_a$-equivariant affinemodification.

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