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.



Non-Ruled Minimal Surfaces
$K^2 >0$ and $S$ non ruled implies that $S$ is of general type. In particular $mK$ is effective for some $m \geq 1$.


Since $H$ is very ample it follows $mKH >0$, that is $KH >0$.


