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Affine pseudo-planes with torus actions

Authors:	Masayoshi Miyanishi Kayo Masuda

Institution:	(1) School of Science and Technology, Kwansei Gakuin University, Sanda, Hyogo 669-1337, Japan;(2) Graduate School of Material Sciences, University of Hyogo, Shosha, Himeji 671-2201, Japan

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 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.

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