Affine surfaces with $AK(S)=\mathbb{C}.$

Abstract

In this paper we give a description of hypersurfaces with trivial ring $AK(S)$, introduced by the second author as following. Let $X$ be an affine variety and let $G(X)$ be the group generated by all ${\mathbb{C}}^{+}$-actions on $X$. Then $AK(X)$ is the subring of all regular $G(X)-$ invariant functions on $X.$ We show that a smooth affine surface $S$ with $AK(S)=\mathbb{C}$ is quasihomogeneous and so may be obtained from a smooth rational projective surface by deleting a divisor of special form, which is called a ``zigzag''. We denote by $A$ the set of all such surfaces, and by $H$ those which have only three components in the zigzag. We prove that for a surface $S\in A$ the following statements are equivalent: 1. $S$ is isomorphic to a hypersurface; 2. $S$ is isomorphic to a hypersurface, defined by equation $xy=p(z)$ in ${\mathbb{C}}^3,$ where $p$ is a polynomial with simple roots only; 3. $S$ admits a fixed-point free ${\mathbb{C}}^{+}$- action; 4. $S\in H.$ Moreover, if $S_1$ belongs to $H,$ and $S_2$ does not, then $S_1\times {\mathbb{C}}^k\ncong S_2\times {\mathbb{C}}^k$ for any $k\in \mathbb{N}$.

23 pages, AMSTeX

References in corpus (1)

• Affine surfaces with isomorphic cylinders