The key point is to show the following: let $f:V\rightarrow W$ be a regular map of varieties over an algebraically closed field $k$; if $V(k)\rightarrow W(k)$ is surjective, then $V(K)\rightarrow W(K)$ is surjective for every algebraically closed field $K$ containing $k$.

We prove this by induction on the dimension of $W$. We may suppose that $W$ is an irreducible closed subvariety of some affine space. Let $P\in W(K)$ be not in the image. We know that $tr.deg.k(P)\leq dimW$. If equality holds, then $P$ and its conjugates under $Aut(K/k)$ are Zariski dense in $W_{K}$, contradicting the fact that $f$ is (obviously) dominant. Hence
 $P\in Z(K)$ for some proper closed irreducible subvariety $Z$ of $W$. Now apply induction to $f^{-1}(Z)\rightarrow Z$ to get a contradiction.