Affine $SL(2)$-surfaces

Let $\mathbb{k}$ be a field of characteristic zero and $X$ be $D//Z_2$, where $D$ is the Danielewski surface given by the equation $2xy-z^2 = 1$. We plan to discuss and prove some properties of the smooth $SL(2)$-surface $X$. We will prove that $SL(2)$-action on $X$ does not extend to four-dimensional affine space, however there exists an equivariant algebraic embedding of $X$ into five-dimensional affine space. Besides, we will give a counter-example to the generalized Cancellation Problem using a construction with $X$ and describe the automorphism group of $X$. The talk is based on (G. Freudenburg. A note on smooth affine $SL(2)$-surfaces. arXiv:2411.15879).