Embeddings of Danielewski surfaces in affine space

Abstract

We construct explicit embeddings of Danielewski surfaces \cite{DubG03} in affine spaces. The equations defining these embeddings are obtained from the $2\times 2$ minors of a matrix attached to a weighted rooted tree $\gamma$. We characterize those surfaces $S_{\gamma }$ with a trivial Makar-Limanov invariant in terms of their associated trees. We prove that every Danielewski surface $S$ with a nontrivial Makar-Limanov invariant admits a closed embedding in an affine space ${\mathbb{A}}_k^n$ in such a way that every ${\mathbb{G}}_{a,k}$-action on $S$ extends to an action on ${\mathbb{A}}^n$ defined by a triangular derivation. We show that a Danielewski surface $S$ with a trivial Makar-Limanov invariant and non-isomorphic to a hypersurface with equation $xz-P(y)=0$ in ${\mathbb{A}}_k^3$ admits nonconjugated algebraically independent ${\mathbb{G}}_{a,k}$-actions.