Danielewski超曲面的分类
Classification(s) of Danielewski hypersurfaces
摘要：
Danielewski超曲面是由$x^ny=Q(x,z)$形式的方程定义的$\mathbb{C}^3$中的超曲面$X_{Q,n}$，其中$n\geq1$和$Q(x,z)$是多项式，使得$Q(0,z)$的次数至少为2。在过去的二十年里，许多作者对它们进行了研究。在本文中，我们给出了它们作为代数簇的分类。我们也给他们的分类的环境空间的自同构。作为推论，我们得到了每一个具有$n\geq2$的Danielewski超曲面$X_{Q,n}$至少允许两个非等价嵌入到$\mathbb{C}^3$中.
英文摘要：
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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