A note on the Jacobian Conjecture

Let $F:\mathbb C^n\to \mathbb C^n$ be a polynomial mapping with non-vanishing Jacobian. If the set $S_F$ of non-properness of $F$ is smooth, then $F$ is a surjective mapping. Moreover, if $S_F$ is connected, then $\chi (S_F) \gt 0.$ Additionally, if $n=2$, then $S_F$ cannot be a curve without self-intersections.