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Let $R$ be a Henselian discrete valuation ring with residue field $k$ of positive characteristic and fraction field $K$ of characteristic zero. Let $\pi:X_R \to \mathrm{Spec}(R)$ be flat, projective family of curves with smooth generic fiber. The special fiber has at most nodal singularities. Then,

1) Considering the morphism $\mathrm{Pic}(X_R) \to \mathrm{Pic}(X_k)$ as a morphism of schemes, are the fibers always irreducible? Are they of dimension $h^1(\mathcal{O}_{X_k})$? (the later dimension count is my guess using a torsor action argument from deformation theory)

2) Can the specialization map $\mathrm{Pic}(X_K) \to \mathrm{Pic}(X_k)$ be interpreted as a morphism of schemes (rather than just as groups)? If so, are the fibers irreducible and of what dimension?

Any reference/hint in this direction will be most welcome.

NB. If necessary assume that $X_R$ is regular.

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    $\begingroup$ What? There is a morphism from the relative Picard stack of $X_k/k$ to the relative Picard stack of $X_R/R$. Are you asking about the group homomorphism from the group of $R$-valued points of the Picard stack (modulo isomorphism) to the group of $k$-valued points? How are you thinking of those groups as schemes? $\endgroup$ – Jason Starr Feb 3 '16 at 17:48
  • $\begingroup$ @JasonStarr We have a natural (pullback) functor from invertible sheaves on $X_R$ to that on $X_k$. Assuming the Picard functors are representable by the corresponding Picard groups, doesnt the Yoneda embedding give us a morphism mentioned in question $1$? $\endgroup$ – user45397 Feb 4 '16 at 9:43
  • $\begingroup$ There is a map from the group of $R$-valued points to the group of $k$-valued points. $\endgroup$ – Jason Starr Feb 4 '16 at 10:58

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