Affine threefolds admitting $G_a$-actions

TL;DR

This paper investigates the structure and singularities of affine threefolds with $ ext{G}_a$-actions, focusing on quotient surfaces, fibers, and actions on higher-dimensional affine varieties.

Contribution

It provides new insights into the structure of affine threefolds with $ ext{G}_a$-actions and explores their quotient morphisms and singular fibers.

Findings

Analysis of quotient surface singularities

Description of singular fibers in quotient morphisms

Study of $ ext{G}_a$ and $ ext{G}_m$-actions on affine 4-folds

Abstract

Affine varieties of dimension greater than two can be explored their structures with the help of fibrations by the affine line or plane and quotient morphisms by $\mathbb{G}_a$-actions. We consider $\mathbb{G}_a$-actions on affine threefolds and discuss the structure and the singularities of the quotient surface as well as the singular fibers of the quotient morphism. Relative $\mathbb{G}_a$ and $\mathbb{G}_m$-actions on $\mathbb{A}^4$ or on an affine 4-fold having $\mathbb{A}^3$-fibrations over $\mathbb{A}^1$ are discussed when the actions leave one variable invariant or the fibration invariant.