Theorem \[irreg\], we may assume that $q(X) = 0$. We may also assume that $0 = h^0(X, K_X + L_{\rm red}) $ ($\ge p_g(X)$). So $X$ is the blow up of an Enriques surface by the classification theory. On the other hand, $\pi_1(X) = (1)$ by Proposition \[treeprop\], a contradiction. This proves the theorem.

Surfaces with negative $\kappa$, Part I : ruled surfaces
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In this section, we prove Conjecture \[mainconj\] for relatively minimal surfaces $X$ of Kodaira dimension $\kappa(X) = - \infty$. By Theorem \[irreg\], we may assume that $q(X) = 0$, so $X$ is a relatively minimal rational surface. If $X= \bf{P}^2$ or $\bf{P}^1 \times
\bf{P}^1$, it is easy to verify that Conjecture \[mainconj\] is true since effective divisor is then nef. We thus assume that $X$ is the Hirzebruch surface $\mathbb{F}_d$ of degree $d \ge 1$ (though, $\mathbb{F}_1$ is not relatively minimal).

We first fix some notations. Let $\pi:\mathbb{F}_d \to
\bf{P}^1$ be the ruling. Let $F$ be a general fibre and $C$ the only negative curve (a cross-section, indeed) on $\mathbb{F}_d$. So $C^2 = -d$.

\[ruled\] Let $X$ be a relative minimal surface of Kodaira dimension $\kappa(X) = - \infty$. Then Conjecture \[mainconj\] holds for effective ${\bold 