Regular subrings of a polynomial ring. II (Q790202)

[For part I see ibid. 17, 329-338 (1980; Zbl 0461.13009).] Let X be a non-singular affine surface over the algebraically closed characteristic zero field k, and suppose there is a morphism p from X to the affine line whose general fibres are affine lines. Then every fibre is (as a cycle) a sum \(n_ 1C_ 1+...+n_ sC_ s\), where \(C_ i\) is an affine line and \(n_ 1>0\). The fibre is singular if \(s\leq 2\), of the first kind if \(n_ i=1\) for some i and of the second kind if \(n_ i\geq 2\) for all i. The author proves that X is dominated by an affine space if and only if p has at most one singular fibre of the second kind. (If A is the coordinate ring of X, this result is a theorem about embedding A in a polynomial ring, whence the paper's title.) - The author further gives some partial results on this embedding problem for A in case the range of p is the projective line.