Let $S$ be a henselian DVR and $X/S$ be a flat and proper curve with $X$ being regular. Under what conditions the specialization map $Pic^0_{X/S}(S)\to Pic^0_{X/S}(Spec(k(s)))$ is surjective? Here $s\in S$ is the closed point. I want to understand the case when $X_s$ is reducible and generic fiber is smooth.
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Since $\dim X_s = 1$, we have $H^2(X_s, \mathcal{O}_{X_s}) = 0$. Therefore, by deformation theory every line bundle on $X_s$ extends to a line bundle on the formal scheme $\widehat{X}$ (completion along $X_s$). By Grothendieck's existence theorem, the restriction (or "formal completion") map $$ {\rm Pic}(X) \longrightarrow {\rm Pic}(\widehat{X}) $$ is an isomorphism. It follows that every line bundle on $X_s$ extends to $X$.
answered Nov 19 at 8:04
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2$\begingroup$ Grothendieck's theorem works if $S$ is complete, not just henselian. I think you need the fact that $\mathrm{Pic}^0_{X/S}$ is representable (and smooth over $S$). $\endgroup$ Commented Nov 19 at 10:07
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$\begingroup$ @LaurentMoret-Bailly good point! Yes, either that or Artin approximation for coherent sheaves. $\endgroup$ Commented Nov 20 at 6:17
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2$\begingroup$ Yes, but for Artin approximation $S$ should be excellent. $\endgroup$ Commented Nov 20 at 8:10