Global sections of a line bundle with negative self intersection

Question

Let $X$ be a smooth projective surface over complex numbers. Consider a line bundle $L$ on $X$ such that it's self intersection $L\cdot L$ is negative. I read that it means that $L$ is rigid and cannot be moved. Does the negative intersection number mean $H^0(L)=0$?

Answer 1

If $X$ is the blowup of another surface $Y$ along a point and $E$ is the exceptional divisor of the blowup then $L = O_X(E)$ has negative self-intersection, and at the same time has a global section. So, the answer is no.