More étale covers of affine spaces in positive characteristic

We prove that every geometrically reduced projective variety of pure dimension $n$ over a field of positive characteristic admits a morphism to projective $n$-space, étale away from the hyperplane $H$ at infinity, which maps a chosen divisor into $H$ and some chosen smooth points not on the divisor to points not in $H$. This improves an earlier result of the author, which was restricted to infinite perfect fields. We also prove a related result that controls the behavior of divisors through the chosen point.