TITLE: Cartier divisors and global sections
QUESTION [5 upvotes]: I have a brief question - I seem to have a vague recollection that if we have a Cartier divisor $D$ on a scheme $X$ , then we can determine whether $D$ is effective by saying whether $\mathcal{O}_X(D)$ has a global section or not. I have tried to prove this fact, but can't seem to do it (is it true?) . If anyone could confirm / deny this I would be very happy, and I would also be glad if someone could give me a reference or a proof of the fact!
Sincerely
Tedar

REPLY [4 votes]: I know this question has been answered long ago... but I feel that the question is settled more completely by the following simple fact:
$\mathcal O_X(D)$ has a global section if and only if $D$ is linearly equivalent to an effective divisor. (see See this MS question)
Given this, is clear that effectivity implies the existence of a global section; while a global section doesn't imply effectivity (because an effective divisor can be linearly equivalent to a noneffective one).