Let $C/K$ be a smooth projective curve, $K$ a field of characteristic $p$, and $q = p^r$. In Silverman (Arithmetic of Elliptic Curves), on page 25, the curve $C^{(q)}$ is defined as the zero set of the ideal generated by $\{ f^{(q)} : f \in I(C) \}$. (Here $f^{(q)}$ is the polynomial $f$ with each coefficient raised to the power $q$.)
$C^{(q)}$ is the image of $C$ under the $q$-power frobenius map. Is $C^{(q)}$ also smooth?
(I ask because I was reading Cor 2.12 where a rational map of smooth curves is split into the Frobenius and a separable map, and I'm wondering if the separable map is a morphism as well.)
Suppose $C$ is given by $f=0$ and so $C^{(q)}$ is given by $g=f^{(q)}=0$, in the notation of Silverman. If $C^{(q)}$ is not smooth at $Q$, then there is a singularity, where all the partial derivatives of $f^{(q)}$ vanish at $Q$. Since the $q$-th power Frobenius is an automorphism, we can find a point $P$ on $C$ such that $P^{(q)}=Q$, and all the partial derivatives of $f$ vanish at $P$ (here we are using the fact that derivatives and $q$-th power Frobenius on coefficients commute). Thus, $C$ would be singular at $P$.