about the spanned divisor of a complex variety

Question

I have this definition:
let $\xi \in H^1(X,O^*)$ a cocycle. We say that $\xi$ is spanned if for every point $x$ in my variety $X$ there exist a section $s \in H^0(X,O(\xi))$ such that $s(x) \neq 0$.
If $\xi$ is spanned i know that the base locus of the complete linear system associated to $[D]=\xi$, named $|D|$ , is base point free. How can i see that $Bs|D|=\emptyset$ ?
I think that $\forall x\in X $ i have to found a divisor $ D^{'}$ such that $x\notin supp(D^{'})$ . How can i construct this $D^{'}$?