Embeddings of Danielewski surfaces in affine space

Abstract

We construct explicit embeddings of Danielewski surfaces [4] 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.