Oh, ok thanks! I think that the formulation was not good. I think you mean that we can take $X = Y$ (regular) and f the absolute Frobenius, because it is flat (Kunz) and df^p = 0 for every function... In the first step of the construction of the splitting, I need to show that the pullback map to $Y =\mathbb{P}(\mathcal{E}^{*})$ is injective. Is that true?

Thanks for the information. This is surprising to me :)

for me it isn't clear that $F$ injective implies that $f$ is open embedding. How to prove it? Is this a general fact?

@JasonStarr: thanks for the argument. In case, $f(x) = 4x^{2}-x$ the jacobian condition ($\det J_{f} = 1$) isn't satisfied.