Smoothness of the Picard group of a smooth curve Let $X$ be a smooth projective curve over $k=\bar{k}$ and denote its Picard group by $\operatorname{Pic}(X)$, with the usual scheme structure coming from the representability of the relative Picard functor.
It's well known that $\operatorname{Pic}(X)$ is smooth of dimension $g$ everywhere in the case of characteristic $0$.
For positive characteristic, Igusa and Serre constructed examples of smooth surfaces presenting singular Picard groups.
What can be said about smooth curves in positive characteristic? Is the Picard group always smooth?

The Picard group splits as the product of $\mathbb{Z}$ and the Jacobian variety of $X$, and so each connected component of $\mbox{Pic}(X)$ is (non-canonically) isomorphic to the Jacobian of $X$ which is smooth.
Edit: To see that the Picard group splits, consider the exact sequence 
$$0\to\mbox{Pic}^0(X)\to\mbox{Pic}(X)\stackrel{\deg}{\to}\mathbb{Z}\to0$$
where $\mbox{Pic}^0(X)=\{\mathcal{O}_X(D):\deg(D)=0\}$. This sequence splits since if $p_0\in X$, we have a section $\mathbb{Z}\to\mbox{Pic}(X)$ where $m\mapsto\mathcal{O}_X(mp_0)$. It is well-known (and most of the time defined this way) that the Jacobian of $X$ is isomorphic to $\mbox{Pic}^0(X)$.
