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

TL;DR

This paper introduces a canonical factorization for quotient morphisms of affine $ ext{G}_a$-varieties, providing an algorithm to compute it and illustrating its application through examples, including the affine 3-space with a specific group action.

Contribution

It establishes a unique sequence of dominant $ ext{G}_a$-equivariant morphisms that factor the quotient morphism, along with an algorithm to determine the associated degree modules.

Findings

The canonical factorization is uniquely determined by degree modules.

An explicit algorithm for computing the factorization is provided.

Examples include the homogeneous (2,5)-action on $ ext{A}^3$.

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.