An organizing principle in the study of the Jacobian Conjecture

Let $\mathrm{\Omega }$ be an irreducible component of the locus of polynomial maps $F:{\mathbb{C}}^n\to {\mathbb{C}}^n$ satisfying $\text{deg}F\le k$ and $\text{det}DF=1$. It is shown that either $\mathrm{\Omega }\subset \text{Aut}({\mathbb{C}}^n)$, as claimed by the Jacobian Conjecture, or the {\em general} $F\in \mathrm{\Omega }$ is not an automorphism.