I am reading a proposition in Prof. Vakil's notes where he shows that if $X$ is a normal and Noetherian scheme, then $\text{div}$ is injective. He opens by saying that if $\text{div} (\mathcal{L},s) = 0$, then $s$ has no poles.
That is fine. What about zeros though? It seems to me that if $s$ has zeros, then the divisor corresponding to it cannot be $0$. Thanks.