Embeddings of Danielewski surfaces

Abstract

A Danielewski surface is defined by a polynomial of the form P = x n z − p(y). Define also the polynomial P = x n z − 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 nonspecial Danielewski surface S, there exist non-equivalent algebraic embeddings of S in C 3 . 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 O(2) on C 4 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 C 3 are found. We generalize their result and show how this leads to an example of two smooth algebraic hypersurfaces in C 3 which are algebraically non-isomorphic but holomorphically isomorphic.

Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space

“…there is no algebraic automorphism of C 3 mapping one image onto the other [20]. In the same paper Freudenburg and Moser show that the constructed embeddings are holomorphically isomorphic using the linearization results of Heinzner and the first author [27].…”

Embeddings of a family of Danielewski hypersurfaces and certain \mathbf{C}^+-actions on \mathbf{C}^3

“…The Euler characteristic of W 1 is 1. V 0 and V 1 are non-singular, algebraically non-isomorphic but analytically isomorphic ( [5]). The Euler characteristic of these surfaces is 2.…”

“…In [8] and [5], inequivalent embeddings are given. In [8], Shpilrain and Yu use the gradients to distinguish the embeddings.…”

“…The examples of [5] are different, because the authors give analytically equivalent polynomials. But the method used was also to find polynomials p and q such that the zero fibers {p = 0} and {q = 0} are isomorphic but the fiber {p = 1} is not isomorphic to the fiber {q = c} for any c ∈ C. Thus the embeddings of the zero fibers are inequivalent.…”

Real and Rational Forms of Certain O2(?)-actions, and a Solution to the Weak Complexification Problem

“…In [10], it was shown by Miki that certain families of non-linearizable actions for k = C restrict to non-linearizable actions for k = R. We look at these actions from a different point of view, and we show that the same construction gives non-linearizable actions over any field of characteristic zero. Our proof uses facts about embeddings of Danielewski surfaces established in [4].…”