Affine surfaces with $AK(S)=\Bbb C.$
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 $\Bbb {C}^+$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 $AK(S)=\Bbb C$ 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 $\Bbb {C}^3 ,$ where $p$ is a polynomial with simple roots only; 3. $S$ admits a fixedpoint free $\Bbb {C}^+$ action; 4. $S\in H.$ Moreover, if $S_1 $ belongs to $H,$ and $S_2$ does not, then $S_1\times \Bbb {C}^k\not\cong S_2\times \Bbb {C}^k$ for any $k\in\Bbb N$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 July 2000
 arXiv:
 arXiv:math/0007022
 Bibcode:
 2000math......7022B
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 23 pages, AMSTeX