Let $X \to A$ be a smooth, proper morphism of schemes, and let $\Omega_{X/A}^1$ denote the relative cotangent sheaf of the total space $X$ as a scheme over $A$, where we assume that $A$ is a one-point space. Suppose $\mathcal{L} \subseteq \Omega_{X/A}^1$ is a invertible subsheaf of the relative cotangent sheaf.

Is $\mathcal{L} \in \operatorname{Pic}_{X/A}(A)$?

In other words, is the line subbundle $\mathcal{L}$ an element of the $A$-points of the relative Picard scheme of $X$ as a scheme over $A$.

Moreover, suppose that $\operatorname{Pic}_{X/A}(A)$ is isomorphic to the Picard group of the fiber $X \times_A k$.

Would this isomiorphism imply that any line bundle of $\operatorname{Pic}_{X/A}(A)$ is a trivial extension of a corresponding line bundle in $\operatorname{Pic}(X \times k)$?


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