Affine threefolds admitting $G_a$-actions

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.

This paper has been withdrawn by the authors due to an error in the proof of Thm.3.1 which is not true in the present version