Cartier divisors and global sections 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

Effective cartier divisors are just closed subschemes which locally are cut out by the vanishing of a non-zero divisor.
I don't know what you mean by $\mathcal{O}_X(D)$, but if you mean $\mathcal{O}_D(D)$, then certainly it need not have a nonconstant global section. Take for example a family of elliptic curves over $\mathbb{A}^1$. Each elliptic curve is a cartier divisor, but being a projective curve, doesn't have any nonconstant global sections.
See, for example, section 9 of http://stacks.math.columbia.edu/download/divisors.pdf
