Let $X$ be a smooth, projective surface (i.e. $2$-dimensional connected variety) over $k=\mathbb{C}$. Denote by $\mathrm{Pic}_{X/k}$ the associated relative Picard scheme. We write $\mathrm{Pic}^0_{X/k}$ for the connected component of the identity.
For a $k$-scheme $T$, how does $\mathrm{Pic}^0_{X/k}(T)$ look like?
I know that for $Y$ a smooth, projective curve we have $\mathrm{Pic}^0_{Y/k}(T)= \{L \in \mathrm{Pic}_{Y/k}(T) | \mathrm{deg}(L|_{X \times t})=0 $ for all $ t \in T\}$ (see e.g. Bosch, Lütkebohmert, Raynaud). I am reading a book by Badescu where he claims that in the situation above we have $\mathrm{Pic}^0_{X/k}(T)= \{L \in \mathrm{Pic}_{X/k}(T) | L|_{X \times t}$ is algebraically equivalent to $\mathcal{O}_X$ for all $t \in T\}$, but he does not prove anything and I can't seem to find a reference for that. It would be nice if anybody knew one.
Now, let us assume that indeed $\mathrm{Pic}^0_{X/k}$ has this form. Can being in $\mathrm{Pic}^0_{X/k}(T)$ be checked on closed points in $T$?
I.e., is
$\mathrm{Pic}^0_{X/k}(T)= \{L \in \mathrm{Pic}_{X/k}(T) | L|_{X \times t}$ is algebraically equivalent to $\mathcal{O}_X$ for all $t \in T$ closed$\}$?