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Étale endomorphisms of algebraic surfaces with $G_m$-actions


Let X be an algebraic surface defined over the complex field \({\vec C}\) and endowed with an \({\vec A}^1_*\)-fibration. As such a surface we have a Platonic \({\vec A}^1_*\)-fiber space, a weighted hypersurface with its singular point deleted off and, more generally, an affine algebraic surface with an unmixed \(G_m\)-action and its fixpoint deleted off. We consider an étale endomorphism \(\varphi : X \to X\) and show that \(\varphi\) is an automorphism in most cases. Of particular interest is the case of a Platonic \({\vec A}^1_*\)-fiber space, for which \(\varphi\) being an automorphism is closely related to the Jacobian Problem for the affine plane \({\vec A}^2\). We also investigate the automorphism group of such surfaces.

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Received January 7, 1999 / Published online October 30, 2000

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Masuda, K., Miyanishi, M. Étale endomorphisms of algebraic surfaces with $G_m$-actions. Math Ann 319, 493–516 (2001).

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  • Singular Point
  • Automorphism Group
  • Algebraic Surface
  • Complex Field
  • Fiber Space