Here's one way of seeing why Piotr's claim is true. Write $g$ for the genus of $X$ and begin with the following observation. If $p_1, \ldots, p_g$ are points on $X$, then the divisor $E=p_1+\ldots+p_g$E=p_1+\cdots+p_g$ has $l(E) = g + 1 - \rho$, where $\rho$ is the rank of the associated Brill–Noether matrix. This matrix is $g\times g$ so for points $p_1, \ldots, p_g$ in general poisiton, we'll have $\rho=g$ and consequently $l(E)=1$. (For a more precise statement, see §7c in Gunning, Lectures on Riemann Surfaces, PUP 1966.) Now take one such $E$ and consider the divisor $D=E-q$. We have $\deg D = g-1 > 0$ and, for generic $q$, $l(D)=l(E)-1=0$.
If $g=0$ or $1$ then it follows easily from Riemann–Roch that $\deg D > 0 \implies l(D)>0$.
