Let $K$ be a field of characteristic $0$, $X_K$ a smooth projective curve over $K$. Denote by $\bar{K}$ the algebraic closure of $K$. The base change morphism $X_{\bar{K}} \to X_K$, induces via the pull-back map, a morphism of the Picard groups, $\mbox{Pic}(X_K) \to \mbox{Pic}(X_{\bar{K}})$. Is this map injective? If not true in general, is there a known condition on $K$ under which this holds true?