About non ruled surfaces

Question

I have this statement about a surface $S$ (complex projective algebraic variety of complex dimension 2).
Take $S$ a non ruled surface. Then the euler characteristic of $S$ satisfies this inequality:
$e(S)\ge0$.
In particular if $S$ is a surface of general type the inequality is strict.
Using noether's formula can i say something about the sign of $X(O_S)$?
$X(O_S)=\frac{1}{12}(K_S^2+e(S))$ is there an additive information about the self intersection af the canonical line bundle?

Answer 1

Under the hypothesis that $S$ is not birational to a ruled surface, you can show that $\chi(O_S)$ is nonnegative too.

The basic point is that $\chi$ is a birational invariant of smooth varieties. Now since $S$ is not birational to a ruled surface, we can contract $(-1)$-curves successively to get a birational morphism $S \rightarrow S'$ where $S'$ is a minimal surface. By birational invariance we have $\chi(O_S)=\chi(O_{S'})$.

But now $e(S') \geq 0$ by the statement you quoted, and also since $S'$ is minimal $K_{S'}$ is nef, so $K_{S'}^2 \geq 0$ too. So we get $\chi(O_S)=\chi(O_{S'}) \geq 0$.