Affine pseudo-planes with torus actions

Abstract

An affine pseudo-plane X is a smooth affine surface defined over ${\Bbb C}$ which is endowed with an ${\Bbb A}^1$-fibration such that every fiber is irreducible and only one fiber is a multiple fiber. If there is a hyperbolic $\Bbb G_m$-action on X and X is an ${\rm ML}_1$-surface, we shall show that the universal covering $\widetilde{X}$ is isomorphic to an affine hypersurface $x^ry=z^d-1$ in the affine 3-space ${\Bbb A}^3$ and X is the quotient of $\widetilde{X}$ by the cyclic group ${\Bbb Z}/d{\Bbb Z}$ via the action $(x,y,z) \mapsto (\zeta x, \zeta^{-r}y, \zeta^az),$ where $r \geqslant 2, d \geqslant 2, 0 < a < d$ and ${\rm gcd}(a,d) =1.$ It is also shown that a ${\Bbb Q}$-homology plane X with $\overline{\kappa}(X)=-\infty$ and a nontrivial $\Bbb G_m$-action is an affine pseudo-plane. The automorphism group ${\rm Aut}\,(X)$ is determined in the last section.