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 $2\times2$ 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.