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.

20 pages

References in corpus (1)

• Locally nilpotent derivations on affine surfaces with a $\C^*$-action