2000/07/05 by Bandman, Tatiana, Makar-Limanov, Leonid
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.math/0007022
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 ∈ 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 C3 , where p is a polynomial with simple roots only; 3. S admits a fixed-point free \Bbb C+- action; 4. S∈ H. Moreover, if S1 belongs to H, and S2 does not, then S1× \Bbb Ck\not≅ S2× \Bbb Ck for any k∈\Bbb N.