  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, etale 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 our earlier result in math.AG/0207150, which was
restricted to infinite perfect fields. We also prove a related result that
controls the behavior of divisors through the chosen point.
