on $X$ and $L$ in Theorem \[main\].

\[treeprop\] Let $X$ be a nonsingular projective surface with $q(X)=0$ and $L$ a nef and big effective ${\bold Q}$-divisor such that $H^0(X,
K_X + L_{\rm red}) = 0$. Then $\chi({{\mathcal{O}}}_X) = 1$, $L_{\rm red}$ is a connected rational tree and $X$ is simply connected.

Note that $p_g(X) \le h^0(X, K_X + L_{\rm red}) = 0$. So $\chi({{\mathcal{O}}}_X) = 1$. Now the proposition follows from Lemma \[treelem\].

The result below is used in the subsequent sections.

\[fibk=1\] Suppose that $X$ is a minimal nonsingular projective surface with Kodaira dimension $\kappa(X) = 1$, $p_g(X) = 0$, and $\pi_1^{\rm alg}(X) = (1)$ (this is true if $\pi_1(X) = (1)$). Let $\pi : X \rightarrow {\bf
P}^1$ be the unique elliptic fibration with $F$ a general fibre. The following statements are true:

$(1)$ $\pi$ has exactly two multiple fibres $F_1, F_2$, and their multiplicities $m_1, m_2$ are coprime. In particular, if $E$ is horizontal then $E . F = m_1m_2 m_3$ ($\ge 6$) for some positive integer $m_3$.

$(2)$ Suppose further that a reduced connected divisor $D$ on $X$ is a rational tree and contains strictly the support of an effective $\Gamma$ of elliptic fibre type. Then 