In the book Lie Algebras of finite and affine type by Roger Carter, in chapter 3, the conjugacy of Cartan subalgebras of a finite-dimensional Lie algebra over $\mathbb{C}$ is established via a technical lemma in algebraic geometry (Corollary 3.11). This makes me wonder whether a slightly more general version of this corollary holds.
Question 1 Let $n$ be a positive integer, $F: \mathbb{C}^n \to \mathbb{C}^n$ be a polynomial mapping, such that the determinant of the Jacobian $\det(J(F))$ never vanishes, is it true that $F(U)$ is Zariski open for any Zariski open set $U$?
More, generally,
Question 2 Does a similar result holds if we replace $\mathbb{C}^n$ by any smooth algebraic variety over $\mathbb{C}$ and $F$ by a regular mapping?
And even more generally,
Question 3 What happens if we further generalise the above question for a general smooth scheme over an algebraically closed field, which might be of positive characteristic?
I think maybe the strategy for proving Corollary 3.11 in Carter's book can be adapted to answer the above questions in the affirmative. But I prefer a more systematic approach to the above questions, thus simplifying the proof of the conjugacy of Cartan subalgebras of finite-dimensional algebras over $\mathbb{C}$.