Classification(s) of Danielewski hypersurfaces

Pierre-Marie Poloni (1, 2)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 Unibas - Université de Bâle = University of Basel = Basel Universität

The Danielewski hypersurfaces are the hypersurfaces $X_{Q,n}$ in $\mathbb{C}^3$ defined by an equation of the form $x^ny=Q(x,z)$ where $n\geq1$ and $Q(x,z)$ is a polynomial such that $Q(0,z)$ is of degree at least two. They were studied by many authors during the last twenty years. In the present article, we give their classification as algebraic varieties. We also give their classification up to automorphism of the ambient space. As a corollary, we obtain that every Danielewski hypersurface $X_{Q,n}$ with $n\geq2$ admits at least two non-equivalent embeddings into $\mathbb{C}^3$.

Mots clés: Danielewski hypersurfaces; equivalent hypersurfaces; Makar-Limanov invariant; non-equivalent embeddings

Domaines: Géométrie algébrique [math.AG]

Pierre-Marie Poloni. Classification(s) of Danielewski hypersurfaces. 2009. ⟨hal-00441601⟩