Assume there is $\varphi\!: \mathbb{P}^2 \to X$, a purely inseparable rational dominant map over a finite field $k$, where $X$ is an absolutely irreducible smooth surface over $k$. Is there a regular surjective map $\psi\!: \mathbb{P}^2 \to X$ over $k$?



Is there a regular surjective map $\psi\!: \mathbb{P}^2 \to X$ over $k$?
Let X be an unirational K3-surface over algebraically closed field. Its Picard lattice has rank 22. Note that map $\psi\_\*\psi^\*: NS X \rightarrow NS X$ is equal to $\deg \psi$ . But $rk NS \mathbb{P}^2= 1$. Thus we obtain a contradiction. 


