Étale endomorphisms of algebraic surfaces with $G_m$-actions

Abstract.

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.