Antiampleness and ampleness of the Frobenius cokernel

We show that if $X$ is a smooth Fano variety containing a line or a conic with respect to $-K_X$, then the Frobenius cokernel $\mathcal B_X:=\mathrm{coker}(\mathcal O_X\to F_* \mathcal O_X)$ is not antiample; using this criteria, we show that the only smooth Fano threefolds with antiample Frobenius cokernel are $\mathbb P^3$ and the quadric threefold (in characteristic $p\neq 2$), thus answering a question raised by Carvajal-Rojas and Patakfalvi. We also show that for any smooth complete intersection $X\subset \mathbb P^n$ of degree $d_1,\dots,d_c$ such that $\sum d_i = n$ or $n-1$, the Frobenius cokernel is not antiample. We also study the kernels of the higher Cartier operators, and show that for $\mathbb P^n$ and quadric hypersurfaces, all the kernels of the higher Cartier operators are antiample, and thus that the full set of kernels of the Cartier operators cannot characterize projective space. Finally, we show that the Frobenius cokernel is ample if and only if the cotangent bundle is ample.