Let $X$ be a Riemann surface and $D=D^+-D^-$ is divisor(e.g. $P-2Q+3R=(P+3R)-(2Q)$), then I want to ask if the common zeros of $f\in H^0(X,D)$ is exactly the support of $D^-$.
By definition, $f\in H^0(X,D)$ means $(f)\geq-D$, hence the support of $D^-$ is contained in the set of common zeros. But if $p\in X$ is a common zero, then does $p$ belong to the support of $D^-$? If not, could someone give a counterexample?
Also, does the similar result hold for general smooth projective curve over a general field $k$?
Thanks in advance.