Classification(s) of Danielewski hypersurfaces

TL;DR

This paper classifies Danielewski hypersurfaces in complex three-space, detailing their algebraic variety structure and automorphism classifications, and shows multiple non-equivalent embeddings for certain cases.

Contribution

It provides a comprehensive classification of Danielewski hypersurfaces as algebraic varieties and up to automorphisms, revealing multiple embeddings into ^3.

Findings

Complete algebraic classification of Danielewski hypersurfaces

Classification up to automorphisms of the ambient space

Existence of multiple non-equivalent embeddings for n2

Abstract

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 $\mathbb{C}^3$.