Consider a scheme $X$ which is: noetherian, regular, integral. All these properties should be enough to ensure the isomorphism: $\operatorname{Cl }(X)\cong \operatorname{Pic (X)}$. If something is missing, please add the conditions you like on $X$. Here it's important to point out that we formally don't have the equality between $L$ and $\mathcal O_X(D)$.
Now fix an invertible sheaf $L$ on $X$, and suppose that $D$ is a (Cartier or Weil, here it doesn't matter) divisor on $X$ such that $\mathcal O_X(D)\cong L$. In other words $\mathcal O_X(D)$ and $L$ induce the same element in $\operatorname{Pic (X)}$.
Can we always find a rational section of $L$ such that $\operatorname{div}(s)=D$?
Glossary:
- $\mathcal O_X(D)$ is the invertible sheaf associated to the divisor $D$.
- A rational section of $L$ is an element $s\in H^0(X,L\otimes K(X))$. Even if $L$ might not have global sections, rational sections always exist.
- $\operatorname{div}(s)$ is the divisor associated to the rational section $s$. This is a standard contruction similar to the construction of divisors on Riemann surfaces associated to meromorphic functions.
Many thanks