JournalscmhVol. 81 , No. 1DOI 10.4171/cmh/42

Embeddings of Danielewski surfaces in affine space

  • Adrien Dubouloz

    Université de Bourgogne, Dijon, France
Embeddings of Danielewski surfaces in affine space cover

Abstract

We construct explicit embeddings of Danielewski surfaces \cite{DubG03} in affine spaces. The equations defining these embeddings are obtained from the 2×22\times2 minors of a matrix attached to a weighted rooted tree γ\gamma. We characterize those surfaces SγS_{\gamma} with a trivial Makar-Limanov invariant in terms of their associated trees. We prove that every Danielewski surface SS with a nontrivial Makar-Limanov invariant admits a closed embedding in an affine space Akn\mathbb{A}_{k}^{n} in such a way that every Ga,k\mathbb{G}_{a,k}-action on SS extends to an action on An\mathbb{A}^{n} defined by a triangular derivation. We show that a Danielewski surface SS with a trivial Makar-Limanov invariant and non-isomorphic to a hypersurface with equation xzP(y)=0xz-P(y)=0 in Ak3\mathbb{A}_{k}^{3} admits nonconjugated algebraically independent Ga,k\mathbb{G}_{a,k}-actions.