TITLE: Dominant map from affine space implies unirationality
QUESTION [8 upvotes]: 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 $X$.

REPLY [6 votes]: The proof in the case when $k$ is a finite field is given in Lemma 11 in Unirationality of Cubic Hypersurfaces by Kollár, published in Journal of the Institute of Mathematics of Jussieu, Volume 1, Issue 3 (2002). But you can also find the article in
https://arxiv.org/abs/math/0005146 
and Lemma 11 is given in page 5.