# Does every relative curve have a Picard scheme?

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.

• Assume $g=1$. Elliptic curves are canonically projective, so by working etale-locally on the base to attain projectivity we see in general that Pic$_{X/S}$ is an fppf sheaf extension $P$ of $\mathbf{Z}$ by an elliptic curve $E$. Let $P_n$ be the fiber over $n$, so $P_1$ is the curve $X$ that is an $E$-torsor. For $n\ne 0$, $[n]:P \rightarrow P$ expresses $P_n$ as the quotient of $P_1=X$ by the free action of the finite loc. free $E[n]$. So equivalently: is the algebraic space $X/E[n]$ a scheme for $n\ne 0$? By SGA3, Exp. V, Thm. 4.1, same that each fibral $E[n]$-orbit is in an open affine. Nov 10, 2014 at 16:19
• It seems to me that if $X$ is normal and locally factorial then the morphism $f: X\to S$ has to be Zariski-locally projective, which suffices to show Pic is a scheme under those assumptions. Indeed, let $s\in S$ be a point and $U\subset X$ an affine in $X$ such that $f^{-1}(S)\cap U$ is non-empty. Then by normality the complement $D$ of $U$ has pure codimension $1$; by local factoriality it is a Cartier divisor on $X$, and by non-emptyness of $f^{-1}(S)\cap U$ it is relatively ample over a neighborhood of $s$. (Note that $D\cap f^{-1}(s)$ is non-empty as $U$ is affine.) Nov 10, 2014 at 19:06
• @Bjorn Poonen: Is there motivation for this question beyond idle curiosity? Part of the power of algebraic spaces is precisely that they permit us not to get too hung up on the problems of effective descent with schemes and so can move on to more compelling geometric issues. Of course, it comes at the cost of not having affine opens around points anymore, but that is a small price to pay to avoid certain unpleasant technical headaches. Raynaud's theorems on Pic being scheme over a dvr are great, but in generality over base schemes does it really matter if Pic is a scheme or algebraic space? Nov 11, 2014 at 4:29
• Assume $g=1$. If $S$ is normal, then $X=\underline{\mathrm{Pic}}^1_{X/S}$ is locally projective over $S$ by the main theorem in the introduction to Raynaud's thesis (LNM 119, also Corollaire VI.2.5). By the first comment of user52824, it follows that $\underline{\mathrm{Pic}}_{X/S}$ is a scheme (in fact each $\underline{\mathrm{Pic}}^n_{X/S}$ is locally projective over $S$). Nov 11, 2014 at 14:24
• Thank you all for your comments. @user52824 regarding algebraic spaces: Yes, I agree with all that you say. The question was related to something I was working on a few years ago; I ended up not needing the answer, but I am still curious to know it. Nov 12, 2014 at 3:46

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.