More precisely:



> 
> Let $X \to S$ be a smooth proper morphism of schemes such that the geometric fibers
>  are integral curves of genus $g$. Must the fppf relative Picard functor
>  $\operatorname{\bf Pic}\_{X/S}$ be representable by a scheme?
> 
> 
> 


If $g \ne 1$, then some integer power of $\omega\_{X/S}$ 
shows that $X \to S$ is projective 
(Remark 2 on p. 252 of 
*Néron models* by Bosch, Lütkebohmert, and Raynaud),
and then $\operatorname{\bf Pic}\_{X/S}$ is a scheme by
Grothendieck, FGA, no. 232, Theorem 3.1.
So the question is really about the case
in which $g=1$ and $X \to S$ is not projective.



Does every relative curve have a Picard scheme?
The answer is yes: $\mathbf{Pic}\_{X/S}$ is representable by a scheme. I will argue that this follows from the SGA 3 result mentioned by user27920 and from Theorem 2 (c) in section 6.6 of *Neron models* (which itself is based on a nonflat descent result due to Raynaud).


**Preliminary reductions:** Since the $g \neq 1$ case is clear, let us assume that $g = 1$. It suffices to work locally on $S$, so standard limit arguments reduce to the case when $S$ is Noetherian and affine. Since $\mathbf{Pic}\_{X/S} = \bigsqcup\_n \mathbf{Pic}^n\_{X/S}$, it suffices to show each $\mathbf{Pic}^n\_{X/S}$ is a scheme. The latter is a torsor under the elliptic curve $E:=\mathbf{Pic}^0\_{X/S}$, so is an $S$-algebraic space of finite presentation. Further limit arguments therefore reduce to the case when $S$ is in addition local.


**The main argument:** We will argue that every finite set of points of $X$ is contained in an open affine. As explained by user27920, it will then follow from SGA 3, V.4.1 that each $\mathbf{Pic}^n\_{X/S}$ is a scheme.


Choose an affine open $Y \subset X$ that meets the closed fiber of $X \rightarrow S$. Since $X \rightarrow S$ is open and $S$ is local, $Y$ automatically meets every fiber. The action map $E \times\_S X \rightarrow X$ (gotten from the identification $X = \mathbf{Pic}^1\_{X/S}$) is open because it is a base change of $E \rightarrow S$, so the image $EY$ of $E \times\_S Y$ is open. Then checking fiberwise, we conclude that $EY = X$. Therefore, the claim about points being in an open affine follows from the result from *Neron models* mentioned earlier, a simplified version of which says:


**Theorem.** Let $S$ be an affine Noetherian scheme, let $E$ be a smooth $S$-group scheme with connected fibers, and let $E \times\_S X \rightarrow X$ be a group action of $E$ on a smooth $S$-scheme $X$. If there is an affine open subscheme $Y \subset X$ such that $EY = X$, then any finite set of points of $X$ is contained in an open affine.


