Let $G$ be a semisimple linear algebraic group, $P$ be a parabolic subgroup and $w$ be an element of the Weyl group of $G$. I want to calculate the Picard group of the Schubert variety $X_P(w):=\overline{BwP/P} \subset G/P$. I'm particularly interested in the case of a maximal parabolic subgroup, but I suppose the general case is not much harder. In that case, I read without proof, that $\text{Pic}(X_P(w)) \cong \mathbb{Z}$ (of course if $w$ is non trivial). I would also hope, that there is a very ample generator in that case?

I have no problems to calculate the divisor class group, which is freely generated by the divisorial Schubert subvarieties. So the problem lies mainly in the non-smooth case.

A good reference on this topic would also be nice.