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

TL;DR

This paper demonstrates that all deformed Koras-Russell threefolds of the first kind can be realized as algebraic quotients of specific locally trivial -actions on SL_2 imes A^1, revealing a new geometric structure.

Contribution

It establishes a novel connection between deformed Koras-Russell threefolds and algebraic quotients of SL_2 imes A^1 by G_a-actions, expanding understanding of their geometric properties.

Findings

Every such threefold is a quotient of a G_a-action on SL_2 imes A^1.

The G_a-actions considered are proper and Zariski locally trivial.

This provides a new perspective on the structure of Koras-Russell threefolds.

Abstract

Every deformed Koras-Russell threefold of the first kind $Y = \left\{ x^{n}z=y^{m}-t^{r} + xh(x,y,t)\right\}$ in $\mathbb{A}^{4}$ is the algebraic quotient of proper Zariski locally trivial $\mathbb{G}_a$-action on $\mathrm{SL}_2 \times \mathbb{A}^1$.