Some (probably very easy) questions on intersection theory on surfaces...
Say $S$ is a smooth projective surface over $\mathbb{C}$ with canonical divisor $K_S$.
If $S$ is not ruled and $H$ is a hyperplane section (for an arbitrary embedding), do we always have that $K_S \cdot H > 0$? I know that $K_S \cdot H \geq 0$, but why would $K_S \cdot H = 0$ be impossible? EDIT: OK, this can happen if $K_S = 0$, of course (see QiL's answer). But if I moreover assume that $K_S^2 > 0$, can we still have $K_S \cdot H = 0$?
If $D \cdot H < 0$ for some divisor $D$ and some hyperplane section $H$, does it follow that $H^0(D,\mathcal{O}_S(D)) = 0$? It seems reasonable but also too simple, so I'm not so sure.