Proper twin-triangular Ga-actions on A^4 are translations

TL;DR

This paper proves that proper twin-triangular additive group actions on affine 3-space over a complex Dedekind domain are equivalent to translations, using geometric quotient analysis instead of invariant ring computations.

Contribution

It establishes a characterization of proper twin-triangular actions as translations, providing a new geometric approach to understanding these group actions.

Findings

Proper twin-triangular actions are translations.

Properness is equivalent to being a translation for these actions.

The approach avoids invariant ring calculations by analyzing geometric quotients.

Abstract

An additive group action on an affine 3 -space over a complex Dedekind domain A is said to be twin-triangular if it is generated by a locally nilpotent derivation of A[y,z,t] of the form rd/dy+p(y)d/dz + q(y)d/dt, where r belongs to A and p,q belong to A[y] . We show that these actions are translations if and only if they are proper. Our approach avoids the computation of rings of invariants and focuses more on the nature of geometric quotients for such actions.