On a result of Miyanishi-Masuda

Abstract

Let $X$ be an affine surface admitting a unique affine ruling and a ${\mathbb{C}}^{\ast }$-action. Assume that the ruling has a unique degenerate fibre and that this fibre is irreducible. In this paper we give a short proof of the following result of Miyanishi and Masuda: the universal covering of $X$ is a hypersurface in the affine 3-space given by the equation $x^{m}y=z^d-1$, where $m>1$.

3 pages