Varieties with ample Frobenius-trace kernel

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 a 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 .