Etale endomorphisms of algebraic surfaces with G(m)-actions

Let X be an algebraic surface defined over the complex field C and endowed with an A(*)(1)-fibration. As such a surface we have a Platonic A(*)(1)-fib er 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 etale endomorphism phi : X --> X and show that phi is an automorphism in most cases. Of particular interest is the case of a Platonic A(*)(1)-fiber space, for which phi being an autom orphism is closely related to the Jacobian Problem for the affine plane A(2 ). We also investigate the automorphism group of such surfaces.