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It is well known that for a proper smooth variety $X$ over an algebaically closed field $k$, the Picard functor $Pic_{X/k}$ is representable by a smooth group scheme over $k$. My question is, when we retain everything except by replacing the base scheme $k$ by an artin ring, say the dual number $k[\epsilon]/\epsilon^2$, is the Picard functor still representable by a smooth group scheme? More specifically, Is there an answer when $X$ is a flat (nontrivial) deformation of an abelian variety over $k[\epsilon]/\epsilon^2$?

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up vote 2 down vote accepted

In FGA, no.232, Thm 3.1, Grothendieck shows that if $f: X\to S$ is flat, projective and finitely presented, with reduced and irreducible geometric fibers, then $\operatorname{Pic}_{X/S}$ is representable by a separated $S$-scheme, locally of finite presentation over $S$.

As for smoothness, unfortunately the "well-known" result you cite is actually not true in general--in particular, Igusa constructed a smooth projective surface (here is the original paper, written in somewhat archaic language) in characteristic $2$ with non-reduced Picard scheme; the example is the quotient of a particular Abelian surface by a fixed-point-free involution. The smoothness result you claim is true in characteristic zero.

Luckily, Theorem 5 of Section 8.4 of Bosch-Lütkebohmert-Raynaud's Neron Models [BLR] answers your question when $X$ is an Abelian $S$-scheme; in this case the Picard functor is representable by another Abelian $S$-scheme, and is in particular smooth.

Just a warning: in the general case (e.g. if $f: X\to S$ has no section, for example), the Picard functor represented might not be what you think it is, but rather the etale or fppf sheafification of the usual Picard functor. (This can happen e.g. if $X$ has no rational points and $S$ is $\operatorname{Spec}(k)$.) Of course an Abelian scheme always admits a section, so you're OK in this situation.

BTW, Chapter 8 of BLR is a great place to learn about this stuff.

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I think that in BLR's book, they say only that, when $X/S$ is a projective abelian scheme, $\mathrm{Pic}^{\tau}_{X/S}$ is representable. For the representability for the whole Picard functor, you need to use Falting-Chai theorem 1.9, which is due to Raynaud..Moreover, in general, $\mathbb{Pic}_{X/S}$ may be not smooth, only the identity componenet is smooth.. – Tong Feb 6 '13 at 23:15
@Tong: You're completely right. – Daniel Litt Feb 6 '13 at 23:23

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