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.

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

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 $X$.