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Let $C$ be an irreducible curve over a field $k$ and let $X$ be a $k$-variety equipped with a morphism $f: X \to C$. Let $X_{k(C)} \to k(C)$ be the generic fibre of this morphism. Under which "reasonable" conditions on $X$, $C$ and/or $f$ (smoothness, properness and so on) will the natural sequence

$$\text{Pic}\,C \to \text{Pic}\,X \to \text{Pic}\,X_{k(C)} \to 0$$

be exact? For example, does this hold if $X$, $C$ and $f$ are smooth and proper?

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  • $\begingroup$ I'm not sure what exactly you're looking for. Do you want the weakest possible conditions to make this true, or just some conditions maybe to make the proof clean? For example, the first thing that pops into my head is to make the assumption at the beginning of II.6 in Hartshorne so that there is a good interpretation of this sequence in terms of divisors. $\endgroup$
    – Matt
    Mar 17, 2012 at 18:13
  • $\begingroup$ Well, I guess condition (*) on page 130 in Hartshorne is definitely something I should impose. I don't necessarily want the weakest possible conditions - rather some "nice" conditions which allow the proof to be clean. Thanks for your comment! $\endgroup$
    – Evariste
    Mar 17, 2012 at 18:29

2 Answers 2

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You have to suppose $X\to C$ flat to avoid empty generic fiber.

Assume $X$ is regular and flat over $C$.

Then $\mathrm{Pic}(X)\to \mathrm{Pic}(X_K)$, where $K=k(C)$, is surjective. Indeed, identifying invertible sheaves (up to isomorphism) to Weil divisors (up to linear equivalence), it is enough to show that any point of codimension $1$ $P$ in $X_K$ extends to a divisor on $X$. It then suffices to take the Zariski closure of $\{ P\}$.

Now let us look at the exactness at middle. An element of $\mathrm{Pic}(X)$ is in the kernel of $\mathrm{Pic}(X)\to \mathrm{Pic}(X_K)$ if and only if it is represented by a Weil divisor on $X$ supported in finitely many closed fibers of $X\to C$:

(1) if $\mathcal L\in \mathrm{Pic}(X)$ is trivial on $X_K$, dividing by a rational section which is a basis on $X_K$, we can suppose that $\mathcal L$ is a subsheaf of $K(X)$ and equal to $O_X$ on an open subset $U$ containning $X_K$. So $\mathcal L=O_X(D)$ for some Cartier divisor $D$ supported in $X\setminus U$. As $F=f(X\setminus U)$ is constructible hence finite, $D$ is supported in $f^{-1}(F)$.

(2) Conversely, a divisor supported in a finite union of closed fibers is clearly trivial on $X_K$.

So the exactness at the middle is equivalent to saying that any vertical divisor is principal. Note that $f(X)$ is open in $C$ and $f(X)$ is regular because $X$ is regular and $X\to f(X)$ is faithfully flat. Now it is enough (and essentially necessary) to suppose the fibers of $X\to C$ are integral because every closed fiber $X_s$ is then a principal divisor (if $s\notin f(X)$, there is nothing to prove; if $s\in f(X)$, then $[s]$ is a principal divisor and so is $[X_s]=f^*[s]$).

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  • $\begingroup$ Dear QiL, thank you very much for your reply. I still have a few questions. (1) In the first part of your reasoning (surjectivity), you mean a codimension 1 point on Xk(C) instead of a closed point, right? (2) The second sentence of your second paragraph sounds very reasonable, but what is the formal argument? (3) The last sentence of your second paragraph sounds very reasonable as well, but I am not sure why you need $C$ to be regular to make things work... Sorry for my incomprehension, and thanks again! $\endgroup$
    – Evariste
    Mar 17, 2012 at 22:37
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    $\begingroup$ @Evariste: (1) Sorry I had the situation of relative curves in mind. I will add some details for (2). The regularity of $C$ is in fact implied by that of $X$. $\endgroup$
    – user18119
    Mar 17, 2012 at 23:33
  • $\begingroup$ Great, thanks a lot! $\endgroup$
    – Evariste
    Mar 18, 2012 at 0:05
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Let $f\colon X\to C$ be a faithfully flat morphism of locally noetherian schemes which is either quasi-compact or locally of finite type, where $C$ is normal and integral with function field $K$. Assume that, for every point $s\in C$ of codimension $1$, the fiber $X_{s}$ is integral. Then the canonical sequence $$ {\rm Pic}\, C\to {\rm Pic}\, X\to {\rm Pic}\, X_{K} $$ is exact. This result is due to Raynaud (see EGA, ${\rm Err}_{\,\rm IV}$, 53, Corollary 21.4.13, p. 361). If, in addition, $X$ is locally factorial, then the right-hand map above is surjective. The following proof of the latter surjectivity was sent to me by Cedric Pepin. By EGA, ${\rm IV}_{4}$, Corollary 21.6.10(ii), the latter map can be identified with the map of divisor class groups ${\frak{Cl}}\, X\to {\frak{Cl}}\, X_{K}$. Thus it suffices to check that every closed and irreducible subscheme $D_{K}$ of codimension 1 in $X_{K}$ extends to a closed and irreducible subscheme $D$ of codimension 1 in $X$. Since ${\rm Spec}\, K\to C$ is quasi-compact, the canonical morphism $D_{K}\to X$ is quasi-compact as well and the schematic closure $D$ of $D_{K}$ in $X$ is defined by EGA 1 (new), Corollary 6.10.6, p. 325. Since $D$ is closed and irreducible of codimension 1 in $X$, the proof is complete. If anyone knows a statement that is more general than the above, please let me know!

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