Proper twin-triangular $\mathbb {G}_{a}$-actions on $\mathbb {A}^{4}$ are translations

Abstract

An additive group action on an affine 3-space over a complex Dedekind domain A is said to be twintriangular if it is generated by a locally nilpotent derivation of A[y, z1, z2] of the form r∂y + p1(y)∂z 1 + p2(y)∂z 2 , where r ∈ A and p1, p2 ∈ 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.

Exotic Ga-quotients of SL$_2 \times \mathbb{A}^1$

“…The proof of Theorem 1 draws as in [3,9] on the study of categorical quotients of certain G a,k -actions on deformed Koras-Russell threefolds in the category of algebraic spaces. The algebraic spaces which come into play are a particular class of "non-separated surfaces with an m-fold curve" which already appeared in the context of the study of proper G a,k -actions on A 4 k in [5,6] and, for some special cases, in [2] and [7] in relation to the Zariski Cancellation problem for threefolds. In many respects, these spaces tend to be natural and necessary replacements in higher dimension of the non-separated curves first considered by Danielewski [1] in its famous counter-example to the Cancellation problem in dimension two, and which became ubiquitous in the study A 1 -fibered affine surfaces after the work of Fieseler [10].…”

“…Another description of the algebraic space quotients. An alternative complementary description of the algebraic space β : Ũ /R → U constructed in subsection 1.2 was given in [5] in a more general context. Since this description is sometimes more convenient to use in practice, let us review it in detail in our particular situation.…”

Proper Ga-actions on ℂ4 preserving a coordinate

“…In his example the geometric quotient of the action is not Hausdorff while for a translation on C n the geometric quotient is isomorphic to C n−1 . Recently Dubouloz, Finston, and Jaradat [12] proved that every triangular action on C n , which is proper (in particular, it is free and has a Hausdoff geometric quotient), is a translation in a suitable coordinate system. Note that every triangular action preserves at least one of coordinates, and 1 In fact, for n ≤ 3 every connected one-dimensional unipotent algebraic subgroup of Cremona group of C n is conjugate to such a translation [39,Corollary 5].…”