Comrades,

Let $S$ be a non-ruled, minimal (smooth projective complex algebraic) surface. Let $K$ be a canonical divisor of $S$ and $H$ a hyperplane section of $S$ (for your favorite embedding).

Suppose I know that $K^2 > 0$. Then supposedly $S$ being non-ruled forces $H.K > 0$. Is this correct or should it only say $H.K \geq 0$? Why is it correct?

This is at the beginning of chapter 9 in beauville's surfaces book. I can say for sure that $H.K \geq 0$ since a surface is ruled iff there exists a non-exceptional curve $C \subseteq S$ satisfying $C.K < 0$.

Thanks in advance.

edited, i said something stupid in original post.