# Classification(s) of Danielewski hypersurfaces

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 $\mathbb{C}^3$.
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Cited literature [10 references]

https://hal.archives-ouvertes.fr/hal-00441601
Contributor : Pierre-Marie Poloni <>
Submitted on : Wednesday, December 16, 2009 - 6:18:19 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM
Long-term archiving on : Thursday, June 17, 2010 - 11:46:13 PM

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### Identifiers

• HAL Id : hal-00441601, version 1
• ARXIV : 0912.3241

### Citation

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

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