Let $T$ be a torus and $X$ a proper smooth curve over characteristic $0$ algebraically closed field $k$.
What is $\text{Pic}(\text{Gr}_{T,X^n})$?
Here $\text{Gr}_{T,X^n}$ is the BD Grassmannian over $X^n$, parametrising $n$-tuples of points in $X$, a $T$-bundle over $X$ and a trivialisation away from those points. n.b. I am asking about all line bundles, not just factorisable ones.
