  We investigate the injectivity of the Frobenius map on thickenings of smooth
varieties in projective space over a field of positive characteristic. We
obtain uniform bounds -- i.e., independent of the characteristic -- on the
thickening that ensures an injective Frobenius map when the projective variety
is a smooth complete intersection or an arbitrary projective embedding of an
elliptic curve. Our bounds are sharp in the case of hypersurfaces, and in the
case of elliptic curves.
