M. Miyanishi generelized the usual Jacobian Conjecture as follows.
Generalized Jacobian Conjecture.
Let V be a normal affine variety. Suppose f:V \to V is an unramified
morphism. Then f is a proper morphism.
After discussing some earlier positive results and some counterexamples to
GJC we will outline the proof of the following result proved jointly with
Theorem. Let V be an irreducible normal affine surface. Assume that V has
at least one singular point which is not a quotient singular point. Then
any unramified self morphism is an