I read in Serre's "Lectures on $N_X(p)$" that when $X$ is a scheme defined over $\mathbb{F}_q$ (a finite field), the *geometric Frobenius* $F: X \mapsto X$ is defined by fixing every element of the topological space of $X$, and acting as $f \mapsto f^q$ on the structure sheaf. As a particular case, if $X$ is affine (say $Spec(\mathbb{F}_{q^n}[x,y])$), then "$F$ is the "standard Frobenius map" (on points): $(u,v) \mapsto (u^q,v^q)$." Why is that ? Doesn't the action on the structure sheaf implies that every element of $\mathbb{F}_{q^n}[x,y]$ is raised to the $q$-th power ? Also, I have seen other definitions of geometric Frobenius. (I have noticed related questions on MO, but the answers confuse me even more.)

Secondly, it is mentioned that *the geometric and arithmetic Frobenius act in the same way on $X(\overline{\mathbb{F}_q})$*. Can anyone explain this in some detail ?

Thanks !