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.