  In the search of a projective analog of Kunz's theorem and a
Frobenius-theoretic analog of Mori--Hartshorne's theorem, we investigate the
positivity of the kernel of the Frobenius trace (equivalently, the negativity
of the cokernel of the Frobenius endomorphism) on a smooth projective variety
over an algebraically closed field of positive characteristic. For instance,
such kernel is ample for projective spaces. Conversely, we show that for
curves, surfaces, and threefolds the Frobenius trace kernel is ample only for
Fano varieties of Picard rank $1$.
