Affine surfaces with $AK(S)=\Bbb C.$

Details Affine surfaces with $AK(S)=\Bbb C.$ Affine surfaces with $AK(S)=\Bbb C.$

2000-07-05

arxiv.org/abs/math/0007022v1

Mathematics

Algebraic Geometry

23 pages, AMSTeX

Scientific paper

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 fixed-point 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 $k\in\Bbb N$.

Affiliated with

Also associated with

No associations

Rating

Affine surfaces with $AK(S)=\Bbb C.$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Affine surfaces with $AK(S)=\Bbb C.$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Affine surfaces with $AK(S)=\Bbb C.$ will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-171758