Line bundle and effective divisor?

Question

$X$ is for me an abelian variety.

I think I have seen this result somewhere but I can not find it anymore:

For any line bundle $\mathcal{L}$, $\dim H^0(X,\mathcal{L})>0$ if an only if $\mathcal{L}\cong \mathcal{O}_X(D)$ for some effective divisor $D$.

Is it true? If yes, can you please give me a reference where I can find it?

Thanks a lot!

Answer 1

This holds for any proper scheme over $k$, since the set of all such effective divisors is in bijection with $(H^0(X,L) - \{0\})/k^*$. See chapter II.7 in Hartshorne's Algebraic geometry, in particular the part about linear series.