Embeddings of Danielewski surfaces

Abstract.

A Danielewski surface is defined by a polynomial of the form P=x ^nz−p(y). Define also the polynomial P ^′=x ^nz−r(x)p(y) where r(x) is a non-constant polynomial of degree ≤n−1 and r(0)=1. We show that, when n≥2 and deg p(y)≥2, the general fibers of P and P ^′ are not isomorphic as algebraic surfaces, but that the zero fibers are isomorphic. Consequently, for every non-special Danielewski surface S, there exist non-equivalent algebraic embeddings of S in ℂ³. Using different methods, we also give non-equivalent embeddings of the surfaces xz=(y ^d _n >−1) for an infinite sequence of integers d _n . We then consider a certain algebraic action of the orthogonal group $\mathcal O(2)$ on ℂ⁴ which was first considered by Schwarz and then studied by Masuda and Petrie, who proved that this action could not be linearized. This was done by comparing the strata of this action to those of the induced tangent space action. Inequivalent embeddings of a certain singular Danielewski surface S in ℂ³ are found. We generalize their result and show how this leads to an example of two smooth algebraic hypersurfaces in ℂ³ which are algebraically non-isomorphic but holomorphically isomorphic.