Suppose $X$ is a variety of dimension $n$ over $k$ and there exists a dominant rational map $\mathbb{A}^N\dashrightarrow X$, where $N$ can be larger than $n$. Is it true that there is a dominant rational map $\mathbb{A}^n\dashrightarrow X$? 


If $k$ is characteristic 0, this is true because we can apply generic smoothness and look at the map of tangent spaces to pick out an $n$-dimensional linear subspace $\mathbb{A}^n$ of $\mathbb{A}^N$ that dominates $X$.


In characteristic $p$, it seems unlikely, but one might worry that every $n$-dimensional linear subspace of $\mathbb{A}^N$ somehow fails to dominate $X$. 



Dominant map from affine space implies unirationality
It seems to me that the following arguement doses not depend on characteristic. Consider a general fiber of $\varphi:\mathbb{A}^N\dashrightarrow X$. Take its closure in $\mathbb{P}^N$. It is a subvariety of dimension $N-n$. Then a general $\mathbb{P}^n$ intersects it in a finite subscheme, so restriction of $f$ to this plane is still dominant.


