If $X$ is a scheme over a scheme $S$ of characteristic $p>0$ and $F:X\to X^{(p)}$ is the relative Frobenius, it is known that there is a canonical connection on the Frobenius pull-back $F^*E$ of a vector bundle (or coherent sheaf) on $X^{(p)}$. This is defined locally via $f\otimes s\mapsto (1\otimes s)\otimes df$
However, I have problems understanding this connection in practical examples. For example if we consider $X=\mathbb{P}^{1}_{k}$ over a field $k$ of characteristic $p>0$, (in which case $X^{(p)}=X=\mathbb{P}^{1}_{k}$) and an invertible sheaf $\mathcal{O}_{\mathbb{P}^{1}_{k}}(n)$, how does this connection map a given section of $F^*\mathcal{O}_{\mathbb{P}^{1}_{k}}(n)$? I would like to see some concrete examples of this computation. Thanks!
I haven't thought concretely about how this connection should look in your example, though I think it is uniquely characterized on local sections by the claim that it annihilates sections of $F^*L$ (for line bundle $L$) which are pulled back from local sections of $L$. This is an example of descent — the connection expresses the structure on a vector bundle that it inherits it by being pulled back under $F$, i.e. sections that "have descent data" are those that are themselves pulled back. Intuitively you should think of pulling back a vector bundle $V$ under a more geometric map $f$ — the pullback has the structure that the fibers $V_x$ and $V_y$ are identified whenever $f(x) = f(y)$, and sections that are preserved by this identification are the same as sections downstairs. Frobenius is a funny map to think about geometrically, in this case this descent data becomes something purely infinitesimal, i.e. a connection, something that doesn't have much of a direct analogue in characteristic zero. This story is developed in Berthelot's texts "$\mathscr{D}$-modules arithmétiques" here and here, under the name Cartier descent, but I don't know where there's much exposition (it's briefly summarized in the paper "Cusps and $\mathscr{D}$-modules" by Ben-Zvi and Nevins here).
Berthelot, Pierre. $\mathscr{D}$-modules arithmétiques. I. Opérateurs différentiels de niveau fini. (French) [Arithmetical $\mathscr{D}$-modules. I. Differential operators of finite level] Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 2, 185-272.
Berthelot, Pierre. $\mathscr{D}$-modules arithmétiques. II. Descente par Frobenius. (French) [Arithmetic -modules. II. Frobenius descent] Mém. Soc. Math. Fr. (N.S.) No. 81 (2000), vi+136 pp.
Ben-Zvi, David; Nevins, Thomas. Cusps and $\mathscr{D}$-modules. J. Amer. Math. Soc. 17 (2004), no. 1, 155–179.
No, that's true about fibers of a pullback — e.g. sections of the trivial rank one bundle are functions, pullback of the trivial bundle is the trivial bundle, but there are more functions upstairs than downstairs (the difference is sections needn't be constant along the fibers — that's exactly what the "descent data" enforces).