Affine pseudo-coverings of algebraic surfaces

An affine pseudo-covering of smooth affine varieties is an étale morphism whose image contains all codimension one points of X. If f splits an étale endomorphism φ:X→X as φ=f g with a dominant morphism , then f and g are affine pseudo-coverings under some additional conditions which are satisfied when X is the affine n-space . Motivated by the Jacobian problem, we consider an affine pseudo-coverings in the case where Y or X is isomorphic to the affine plane .